topic_name stringclasses 721
values | problem stringlengths 4 1.36k ⌀ | hints/solutions stringlengths 5 3.3k |
|---|---|---|
307 | Solve for $k$, $ \dfrac{8}{2k} = -\dfrac{10}{4k} + \dfrac{5k - 10}{k} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2k$ $4k$ and $k$ The common denominator is $4k$ To get $4k$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{8}{2k} \times \dfrac{2}{2} = \dfrac{16}{4k} $ The denominator of th... |
307 | Solve for $x$, $ -\dfrac{5}{8x + 16} = \dfrac{7}{10x + 20} + \dfrac{x + 6}{2x + 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8x + 16$ $10x + 20$ and $2x + 4$ The common denominator is $40x + 80$ To get $40x + 80$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{5}{8x + 16} \times \dfrac{5}{5} = -\df... |
307 | Solve for $t$, $ -\dfrac{5t + 2}{2t + 5} = -\dfrac{8}{2t + 5} + \dfrac{1}{6t + 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2t + 5$ $2t + 5$ and $6t + 15$ The common denominator is $6t + 15$ To get $6t + 15$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{5t + 2}{2t + 5} \times \dfrac{3}{3} = -\df... |
307 | Solve for $p$, $ \dfrac{p - 1}{16p} = -\dfrac{6}{8p} - \dfrac{7}{8p} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16p$ $8p$ and $8p$ The common denominator is $16p$ The denominator of the first term is already $16p$ , so we don't need to change it. To get $16p$ in the denominator of the second term, multiply it by $... |
307 | Solve for $z$, $ \dfrac{3}{4z^2} = -\dfrac{4}{2z^2} - \dfrac{4z - 6}{2z^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4z^2$ $2z^2$ and $2z^2$ The common denominator is $4z^2$ The denominator of the first term is already $4z^2$ , so we don't need to change it. To get $4z^2$ in the denominator of the second term, multiply... |
307 | Solve for $z$, $ \dfrac{10}{8z^2} = \dfrac{z - 3}{8z^2} + \dfrac{10}{16z^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8z^2$ $8z^2$ and $16z^2$ The common denominator is $16z^2$ To get $16z^2$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{10}{8z^2} \times \dfrac{2}{2} = \dfrac{20}{16z^2} $ T... |
307 | Solve for $t$, $ -\dfrac{9}{3t} = \dfrac{5}{3t} + \dfrac{3t + 8}{15t} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3t$ $3t$ and $15t$ The common denominator is $15t$ To get $15t$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{9}{3t} \times \dfrac{5}{5} = -\dfrac{45}{15t} $ To get $15t$ i... |
307 | Solve for $n$, $ \dfrac{n + 7}{5n - 1} = \dfrac{3}{5n - 1} + \dfrac{8}{25n - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5n - 1$ $5n - 1$ and $25n - 5$ The common denominator is $25n - 5$ To get $25n - 5$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{n + 7}{5n - 1} \times \dfrac{5}{5} = \dfrac... |
307 | Solve for $z$, $ \dfrac{4}{4z + 20} = -\dfrac{z - 10}{z + 5} - \dfrac{8}{2z + 10} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4z + 20$ $z + 5$ and $2z + 10$ The common denominator is $4z + 20$ The denominator of the first term is already $4z + 20$ , so we don't need to change it. To get $4z + 20$ in the denominator of the secon... |
307 | Solve for $a$, $ \dfrac{10}{4a^3} = -\dfrac{5a + 9}{4a^3} - \dfrac{5}{4a^3} $ | If we multiply both sides of the equation by $4a^3$ , we get: $ 10 = -5a - 9 - 5$ $ 10 = -5a - 14$ $ 24 = -5a $ $ a = -\dfrac{24}{5}$ |
307 | Solve for $k$, $ -\dfrac{6}{5k + 5} = \dfrac{5k - 7}{10k + 10} + \dfrac{6}{25k + 25} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5k + 5$ $10k + 10$ and $25k + 25$ The common denominator is $50k + 50$ To get $50k + 50$ in the denominator of the first term, multiply it by $\frac{10}{10}$ $ -\dfrac{6}{5k + 5} \times \dfrac{10}{10} = ... |
307 | Solve for $p$, $ -\dfrac{3}{2p} = \dfrac{6}{p} - \dfrac{4p + 5}{p} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2p$ $p$ and $p$ The common denominator is $2p$ The denominator of the first term is already $2p$ , so we don't need to change it. To get $2p$ in the denominator of the second term, multiply it by $\frac{... |
307 | Solve for $t$, $ -\dfrac{4t - 6}{3t^3} = \dfrac{1}{12t^3} + \dfrac{5}{3t^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3t^3$ $12t^3$ and $3t^3$ The common denominator is $12t^3$ To get $12t^3$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{4t - 6}{3t^3} \times \dfrac{4}{4} = -\dfrac{16t - 24... |
307 | Solve for $q$, $ -\dfrac{9}{5q + 5} = -\dfrac{q - 8}{5q + 5} - \dfrac{7}{q + 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5q + 5$ $5q + 5$ and $q + 1$ The common denominator is $5q + 5$ The denominator of the first term is already $5q + 5$ , so we don't need to change it. The denominator of the second term is already $5q + ... |
307 | Solve for $p$, $ -\dfrac{6}{3p} = \dfrac{p + 4}{3p} + \dfrac{1}{3p} $ | If we multiply both sides of the equation by $3p$ , we get: $ -6 = p + 4 + 1$ $ -6 = p + 5$ $ -11 = p $ $ p = -11$ |
307 | Solve for $y$, $ -\dfrac{8}{y + 5} = -\dfrac{1}{y + 5} - \dfrac{4y + 9}{3y + 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $y + 5$ $y + 5$ and $3y + 15$ The common denominator is $3y + 15$ To get $3y + 15$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{8}{y + 5} \times \dfrac{3}{3} = -\dfrac{24}{... |
307 | Solve for $r$, $ -\dfrac{r - 1}{6r} = \dfrac{8}{6r} + \dfrac{1}{6r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $6r$ $6r$ and $6r$ The common denominator is $6r$ The denominator of the first term is already $6r$ , so we don't need to change it. The denominator of the second term is already $6r$ , so we don't need t... |
307 | Solve for $t$, $ \dfrac{5}{2t} = -\dfrac{8}{2t} - \dfrac{3t - 6}{2t} $ | If we multiply both sides of the equation by $2t$ , we get: $ 5 = -8 - 3t + 6$ $ 5 = -3t - 2$ $ 7 = -3t $ $ t = -\dfrac{7}{3}$ |
307 | Solve for $n$, $ -\dfrac{n - 2}{n + 3} = \dfrac{1}{3n + 9} - \dfrac{3}{n + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $n + 3$ $3n + 9$ and $n + 3$ The common denominator is $3n + 9$ To get $3n + 9$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{n - 2}{n + 3} \times \dfrac{3}{3} = -\dfrac{3n ... |
307 | Solve for $t$, $ \dfrac{10}{9t + 12} = -\dfrac{1}{6t + 8} + \dfrac{2t + 1}{15t + 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $9t + 12$ $6t + 8$ and $15t + 20$ The common denominator is $90t + 120$ To get $90t + 120$ in the denominator of the first term, multiply it by $\frac{10}{10}$ $ \dfrac{10}{9t + 12} \times \dfrac{10}{10} ... |
307 | Solve for $r$, $ \dfrac{8}{5r + 1} = -\dfrac{4}{20r + 4} + \dfrac{r}{15r + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5r + 1$ $20r + 4$ and $15r + 3$ The common denominator is $60r + 12$ To get $60r + 12$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ \dfrac{8}{5r + 1} \times \dfrac{12}{12} = \df... |
307 | Solve for $t$, $ -\dfrac{5}{6t} = \dfrac{t + 5}{2t} + \dfrac{1}{4t} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $6t$ $2t$ and $4t$ The common denominator is $12t$ To get $12t$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{5}{6t} \times \dfrac{2}{2} = -\dfrac{10}{12t} $ To get $12t$ in... |
307 | Solve for $a$, $ -\dfrac{5a + 5}{20a - 12} = -\dfrac{1}{20a - 12} - \dfrac{6}{10a - 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $20a - 12$ $20a - 12$ and $10a - 6$ The common denominator is $20a - 12$ The denominator of the first term is already $20a - 12$ , so we don't need to change it. The denominator of the second term is alre... |
307 | Solve for $p$, $ -\dfrac{10}{3p - 5} = \dfrac{5p + 1}{9p - 15} - \dfrac{7}{3p - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3p - 5$ $9p - 15$ and $3p - 5$ The common denominator is $9p - 15$ To get $9p - 15$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{10}{3p - 5} \times \dfrac{3}{3} = -\dfrac{... |
307 | Solve for $x$, $ -\dfrac{7}{2x - 2} = -\dfrac{x}{3x - 3} + \dfrac{3}{4x - 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2x - 2$ $3x - 3$ and $4x - 4$ The common denominator is $12x - 12$ To get $12x - 12$ in the denominator of the first term, multiply it by $\frac{6}{6}$ $ -\dfrac{7}{2x - 2} \times \dfrac{6}{6} = -\dfrac{... |
307 | Solve for $p$, $ -\dfrac{10}{10p^3} = \dfrac{5p - 6}{10p^3} - \dfrac{4}{2p^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10p^3$ $10p^3$ and $2p^3$ The common denominator is $10p^3$ The denominator of the first term is already $10p^3$ , so we don't need to change it. The denominator of the second term is already $10p^3$ , s... |
307 | Solve for $r$, $ \dfrac{6}{8r} = \dfrac{5r + 8}{20r} - \dfrac{5}{4r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8r$ $20r$ and $4r$ The common denominator is $40r$ To get $40r$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{6}{8r} \times \dfrac{5}{5} = \dfrac{30}{40r} $ To get $40r$ in ... |
307 | Solve for $r$, $ \dfrac{r - 5}{2r - 2} = -\dfrac{5}{4r - 4} + \dfrac{4}{5r - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2r - 2$ $4r - 4$ and $5r - 5$ The common denominator is $20r - 20$ To get $20r - 20$ in the denominator of the first term, multiply it by $\frac{10}{10}$ $ \dfrac{r - 5}{2r - 2} \times \dfrac{10}{10} = \... |
307 | Solve for $t$, $ \dfrac{10}{3t + 12} = -\dfrac{5t - 5}{t + 4} - \dfrac{4}{5t + 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3t + 12$ $t + 4$ and $5t + 20$ The common denominator is $15t + 60$ To get $15t + 60$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{10}{3t + 12} \times \dfrac{5}{5} = \dfrac... |
307 | Solve for $k$, $ -\dfrac{5k + 7}{4k - 12} = \dfrac{10}{k - 3} + \dfrac{9}{k - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4k - 12$ $k - 3$ and $k - 3$ The common denominator is $4k - 12$ The denominator of the first term is already $4k - 12$ , so we don't need to change it. To get $4k - 12$ in the denominator of the second ... |
307 | Solve for $y$, $ -\dfrac{3y - 3}{y - 4} = -\dfrac{5}{3y - 12} + \dfrac{1}{5y - 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $y - 4$ $3y - 12$ and $5y - 20$ The common denominator is $15y - 60$ To get $15y - 60$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ -\dfrac{3y - 3}{y - 4} \times \dfrac{15}{15} =... |
307 | Solve for $a$, $ \dfrac{8}{a + 5} = -\dfrac{2}{a + 5} + \dfrac{5a - 4}{a + 5} $ | If we multiply both sides of the equation by $a + 5$ , we get: $ 8 = -2 + 5a - 4$ $ 8 = 5a - 6$ $ 14 = 5a $ $ a = \dfrac{14}{5}$ |
307 | Solve for $k$, $ -\dfrac{4}{k + 2} = -\dfrac{5}{k + 2} + \dfrac{k + 1}{k + 2} $ | If we multiply both sides of the equation by $k + 2$ , we get: $ -4 = -5 + k + 1$ $ -4 = k - 4$ $ 0 = k $ $ k = 0$ |
307 | Solve for $z$, $ \dfrac{z - 7}{z} = \dfrac{3}{z} - \dfrac{1}{3z} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z$ $z$ and $3z$ The common denominator is $3z$ To get $3z$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{z - 7}{z} \times \dfrac{3}{3} = \dfrac{3z - 21}{3z} $ To get $3z$ in... |
307 | Solve for $x$, $ \dfrac{x + 3}{5x + 10} = -\dfrac{6}{4x + 8} + \dfrac{10}{x + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5x + 10$ $4x + 8$ and $x + 2$ The common denominator is $20x + 40$ To get $20x + 40$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{x + 3}{5x + 10} \times \dfrac{4}{4} = \dfr... |
307 | Solve for $p$, $ -\dfrac{6}{2p + 5} = -\dfrac{p + 2}{8p + 20} + \dfrac{1}{2p + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2p + 5$ $8p + 20$ and $2p + 5$ The common denominator is $8p + 20$ To get $8p + 20$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{6}{2p + 5} \times \dfrac{4}{4} = -\dfrac{2... |
307 | Solve for $y$, $ \dfrac{6}{5y + 10} = \dfrac{3y - 6}{3y + 6} - \dfrac{1}{y + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5y + 10$ $3y + 6$ and $y + 2$ The common denominator is $15y + 30$ To get $15y + 30$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{6}{5y + 10} \times \dfrac{3}{3} = \dfrac{1... |
307 | Solve for $p$, $ -\dfrac{10}{3p - 9} = \dfrac{5p - 1}{2p - 6} + \dfrac{3}{p - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3p - 9$ $2p - 6$ and $p - 3$ The common denominator is $6p - 18$ To get $6p - 18$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{10}{3p - 9} \times \dfrac{2}{2} = -\dfrac{20... |
307 | Solve for $z$, $ -\dfrac{z + 4}{8z - 20} = \dfrac{10}{2z - 5} + \dfrac{4}{2z - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8z - 20$ $2z - 5$ and $2z - 5$ The common denominator is $8z - 20$ The denominator of the first term is already $8z - 20$ , so we don't need to change it. To get $8z - 20$ in the denominator of the secon... |
307 | Solve for $r$, $ \dfrac{r + 7}{5r - 20} = \dfrac{5}{3r - 12} + \dfrac{6}{4r - 16} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5r - 20$ $3r - 12$ and $4r - 16$ The common denominator is $60r - 240$ To get $60r - 240$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ \dfrac{r + 7}{5r - 20} \times \dfrac{12}{1... |
307 | Solve for $n$, $ -\dfrac{3n - 1}{n^2} = -\dfrac{9}{4n^2} - \dfrac{1}{n^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $n^2$ $4n^2$ and $n^2$ The common denominator is $4n^2$ To get $4n^2$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{3n - 1}{n^2} \times \dfrac{4}{4} = -\dfrac{12n - 4}{4n^2}... |
307 | Solve for $t$, $ -\dfrac{1}{5t^3} = \dfrac{t - 9}{5t^3} - \dfrac{5}{5t^3} $ | If we multiply both sides of the equation by $5t^3$ , we get: $ -1 = t - 9 - 5$ $ -1 = t - 14$ $ 13 = t $ $ t = 13$ |
307 | Solve for $n$, $ \dfrac{8}{4n + 4} = -\dfrac{3}{5n + 5} + \dfrac{n - 10}{n + 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4n + 4$ $5n + 5$ and $n + 1$ The common denominator is $20n + 20$ To get $20n + 20$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{8}{4n + 4} \times \dfrac{5}{5} = \dfrac{40}... |
307 | Solve for $x$, $ -\dfrac{6}{5x^2} = -\dfrac{3}{25x^2} - \dfrac{x + 10}{5x^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5x^2$ $25x^2$ and $5x^2$ The common denominator is $25x^2$ To get $25x^2$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{6}{5x^2} \times \dfrac{5}{5} = -\dfrac{30}{25x^2} $ ... |
307 | Solve for $n$, $ -\dfrac{7}{n + 5} = \dfrac{3}{n + 5} + \dfrac{3n + 10}{n + 5} $ | If we multiply both sides of the equation by $n + 5$ , we get: $ -7 = 3 + 3n + 10$ $ -7 = 3n + 13$ $ -20 = 3n $ $ n = -\dfrac{20}{3}$ |
307 | Solve for $r$, $ \dfrac{2}{16r} = -\dfrac{8}{20r} - \dfrac{4r - 1}{4r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16r$ $20r$ and $4r$ The common denominator is $80r$ To get $80r$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{2}{16r} \times \dfrac{5}{5} = \dfrac{10}{80r} $ To get $80r$ i... |
307 | Solve for $a$, $ -\dfrac{7}{a + 2} = \dfrac{8}{a + 2} - \dfrac{a - 10}{a + 2} $ | If we multiply both sides of the equation by $a + 2$ , we get: $ -7 = 8 - a + 10$ $ -7 = -a + 18$ $ -25 = -a $ $ a = 25$ |
307 | Solve for $z$, $ -\dfrac{4z + 3}{3z + 2} = \dfrac{3}{3z + 2} - \dfrac{4}{12z + 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3z + 2$ $3z + 2$ and $12z + 8$ The common denominator is $12z + 8$ To get $12z + 8$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{4z + 3}{3z + 2} \times \dfrac{4}{4} = -\df... |
307 | Solve for $k$, $ \dfrac{6}{6k + 6} = \dfrac{k - 7}{3k + 3} - \dfrac{4}{3k + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $6k + 6$ $3k + 3$ and $3k + 3$ The common denominator is $6k + 6$ The denominator of the first term is already $6k + 6$ , so we don't need to change it. To get $6k + 6$ in the denominator of the second te... |
307 | Solve for $p$, $ -\dfrac{5p + 5}{2p} = -\dfrac{1}{2p} + \dfrac{7}{2p} $ | If we multiply both sides of the equation by $2p$ , we get: $ -5p - 5 = -1 + 7$ $ -5p - 5 = 6$ $ -5p = 11 $ $ p = -\dfrac{11}{5}$ |
307 | Solve for $y$, $ \dfrac{y - 9}{5y} = -\dfrac{2}{5y} + \dfrac{10}{5y} $ | If we multiply both sides of the equation by $5y$ , we get: $ y - 9 = -2 + 10$ $ y - 9 = 8$ $ y = 17 $ |
307 | Solve for $z$, $ \dfrac{1}{10z - 6} = \dfrac{5}{10z - 6} - \dfrac{3z + 7}{10z - 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10z - 6$ $10z - 6$ and $10z - 6$ The common denominator is $10z - 6$ The denominator of the first term is already $10z - 6$ , so we don't need to change it. The denominator of the second term is already ... |
307 | Solve for $r$, $ -\dfrac{1}{5r - 5} = \dfrac{6}{r - 1} + \dfrac{4r - 10}{2r - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5r - 5$ $r - 1$ and $2r - 2$ The common denominator is $10r - 10$ To get $10r - 10$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{1}{5r - 5} \times \dfrac{2}{2} = -\dfrac{2... |
307 | Solve for $n$, $ \dfrac{3n - 3}{n - 2} = -\dfrac{4}{4n - 8} + \dfrac{2}{n - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $n - 2$ $4n - 8$ and $n - 2$ The common denominator is $4n - 8$ To get $4n - 8$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{3n - 3}{n - 2} \times \dfrac{4}{4} = \dfrac{12n ... |
307 | Solve for $k$, $ \dfrac{k + 1}{k + 3} = -\dfrac{5}{k + 3} + \dfrac{7}{2k + 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $k + 3$ $k + 3$ and $2k + 6$ The common denominator is $2k + 6$ To get $2k + 6$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{k + 1}{k + 3} \times \dfrac{2}{2} = \dfrac{2k + ... |
307 | Solve for $p$, $ -\dfrac{5}{3p} = -\dfrac{6}{3p} + \dfrac{p + 2}{3p} $ | If we multiply both sides of the equation by $3p$ , we get: $ -5 = -6 + p + 2$ $ -5 = p - 4$ $ -1 = p $ $ p = -1$ |
307 | Solve for $n$, $ -\dfrac{8}{12n - 12} = \dfrac{n - 3}{15n - 15} - \dfrac{2}{15n - 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12n - 12$ $15n - 15$ and $15n - 15$ The common denominator is $60n - 60$ To get $60n - 60$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{8}{12n - 12} \times \dfrac{5}{5} = ... |
307 | Solve for $t$, $ \dfrac{1}{5t^3} = -\dfrac{2}{t^3} - \dfrac{4t - 9}{2t^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5t^3$ $t^3$ and $2t^3$ The common denominator is $10t^3$ To get $10t^3$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{1}{5t^3} \times \dfrac{2}{2} = \dfrac{2}{10t^3} $ To ge... |
307 | Solve for $q$, $ -\dfrac{5q + 5}{25q + 20} = \dfrac{10}{5q + 4} - \dfrac{9}{5q + 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25q + 20$ $5q + 4$ and $5q + 4$ The common denominator is $25q + 20$ The denominator of the first term is already $25q + 20$ , so we don't need to change it. To get $25q + 20$ in the denominator of the s... |
307 | Solve for $k$, $ \dfrac{2k - 9}{k + 3} = -\dfrac{10}{4k + 12} + \dfrac{3}{k + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $k + 3$ $4k + 12$ and $k + 3$ The common denominator is $4k + 12$ To get $4k + 12$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{2k - 9}{k + 3} \times \dfrac{4}{4} = \dfrac{8... |
307 | Solve for $t$, $ -\dfrac{8}{2t + 5} = \dfrac{5}{2t + 5} - \dfrac{2t + 1}{2t + 5} $ | If we multiply both sides of the equation by $2t + 5$ , we get: $ -8 = 5 - 2t - 1$ $ -8 = -2t + 4$ $ -12 = -2t $ $ t = 6$ |
307 | Solve for $n$, $ \dfrac{6}{8n} = -\dfrac{n - 8}{2n} - \dfrac{8}{6n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8n$ $2n$ and $6n$ The common denominator is $24n$ To get $24n$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{6}{8n} \times \dfrac{3}{3} = \dfrac{18}{24n} $ To get $24n$ in t... |
307 | Solve for $r$, $ -\dfrac{4}{4r + 10} = -\dfrac{r - 8}{2r + 5} + \dfrac{1}{2r + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4r + 10$ $2r + 5$ and $2r + 5$ The common denominator is $4r + 10$ The denominator of the first term is already $4r + 10$ , so we don't need to change it. To get $4r + 10$ in the denominator of the secon... |
307 | Solve for $a$, $ \dfrac{a + 8}{12a - 8} = \dfrac{6}{9a - 6} + \dfrac{5}{12a - 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12a - 8$ $9a - 6$ and $12a - 8$ The common denominator is $36a - 24$ To get $36a - 24$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{a + 8}{12a - 8} \times \dfrac{3}{3} = \d... |
307 | Solve for $r$, $ \dfrac{r + 7}{2r + 1} = -\dfrac{6}{10r + 5} + \dfrac{8}{6r + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2r + 1$ $10r + 5$ and $6r + 3$ The common denominator is $30r + 15$ To get $30r + 15$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ \dfrac{r + 7}{2r + 1} \times \dfrac{15}{15} = ... |
307 | Solve for $n$, $ \dfrac{5}{2n - 3} = -\dfrac{4n - 10}{2n - 3} + \dfrac{2}{2n - 3} $ | If we multiply both sides of the equation by $2n - 3$ , we get: $ 5 = -4n + 10 + 2$ $ 5 = -4n + 12$ $ -7 = -4n $ $ n = \dfrac{7}{4}$ |
307 | Solve for $p$, $ -\dfrac{5p + 9}{3p^3} = \dfrac{6}{3p^3} - \dfrac{10}{6p^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3p^3$ $3p^3$ and $6p^3$ The common denominator is $6p^3$ To get $6p^3$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{5p + 9}{3p^3} \times \dfrac{2}{2} = -\dfrac{10p + 18}{6... |
307 | Solve for $y$, $ \dfrac{2}{4y} = \dfrac{3}{8y} + \dfrac{2y + 10}{4y} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4y$ $8y$ and $4y$ The common denominator is $8y$ To get $8y$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{2}{4y} \times \dfrac{2}{2} = \dfrac{4}{8y} $ The denominator of th... |
307 | Solve for $a$, $ \dfrac{5a - 1}{a} = \dfrac{1}{3a} + \dfrac{9}{a} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $a$ $3a$ and $a$ The common denominator is $3a$ To get $3a$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5a - 1}{a} \times \dfrac{3}{3} = \dfrac{15a - 3}{3a} $ The denominat... |
307 | Solve for $y$, $ -\dfrac{9}{y + 5} = \dfrac{5}{4y + 20} - \dfrac{4y - 3}{5y + 25} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $y + 5$ $4y + 20$ and $5y + 25$ The common denominator is $20y + 100$ To get $20y + 100$ in the denominator of the first term, multiply it by $\frac{20}{20}$ $ -\dfrac{9}{y + 5} \times \dfrac{20}{20} = -\... |
307 | Solve for $y$, $ -\dfrac{8}{3y^3} = \dfrac{5y - 9}{3y^3} - \dfrac{10}{3y^3} $ | If we multiply both sides of the equation by $3y^3$ , we get: $ -8 = 5y - 9 - 10$ $ -8 = 5y - 19$ $ 11 = 5y $ $ y = \dfrac{11}{5}$ |
307 | Solve for $n$, $ \dfrac{n - 4}{5n^3} = \dfrac{6}{n^3} - \dfrac{5}{n^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5n^3$ $n^3$ and $n^3$ The common denominator is $5n^3$ The denominator of the first term is already $5n^3$ , so we don't need to change it. To get $5n^3$ in the denominator of the second term, multiply i... |
307 | Solve for $q$, $ -\dfrac{10}{9q + 15} = -\dfrac{q + 5}{12q + 20} - \dfrac{8}{3q + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $9q + 15$ $12q + 20$ and $3q + 5$ The common denominator is $36q + 60$ To get $36q + 60$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{10}{9q + 15} \times \dfrac{4}{4} = -\d... |
307 | Solve for $x$, $ \dfrac{9}{15x + 6} = -\dfrac{10}{20x + 8} - \dfrac{3x - 1}{5x + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15x + 6$ $20x + 8$ and $5x + 2$ The common denominator is $60x + 24$ To get $60x + 24$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{9}{15x + 6} \times \dfrac{4}{4} = \dfrac... |
307 | Solve for $r$, $ \dfrac{7}{r} = -\dfrac{9}{3r} - \dfrac{4r - 7}{r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $r$ $3r$ and $r$ The common denominator is $3r$ To get $3r$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{7}{r} \times \dfrac{3}{3} = \dfrac{21}{3r} $ The denominator of the ... |
307 | Solve for $k$, $ -\dfrac{5}{k^2} = -\dfrac{6}{4k^2} - \dfrac{5k + 2}{k^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $k^2$ $4k^2$ and $k^2$ The common denominator is $4k^2$ To get $4k^2$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{5}{k^2} \times \dfrac{4}{4} = -\dfrac{20}{4k^2} $ The den... |
307 | Solve for $q$, $ -\dfrac{5}{15q - 5} = \dfrac{9}{3q - 1} - \dfrac{4q + 5}{3q - 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15q - 5$ $3q - 1$ and $3q - 1$ The common denominator is $15q - 5$ The denominator of the first term is already $15q - 5$ , so we don't need to change it. To get $15q - 5$ in the denominator of the secon... |
307 | Solve for $r$, $ \dfrac{2}{25r} = \dfrac{3}{15r} + \dfrac{2r + 8}{5r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25r$ $15r$ and $5r$ The common denominator is $75r$ To get $75r$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{2}{25r} \times \dfrac{3}{3} = \dfrac{6}{75r} $ To get $75r$ in... |
307 | Solve for $t$, $ -\dfrac{2t}{3t - 9} = \dfrac{1}{t - 3} - \dfrac{4}{t - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3t - 9$ $t - 3$ and $t - 3$ The common denominator is $3t - 9$ The denominator of the first term is already $3t - 9$ , so we don't need to change it. To get $3t - 9$ in the denominator of the second term... |
307 | Solve for $t$, $ \dfrac{3t - 9}{5t - 15} = -\dfrac{6}{t - 3} + \dfrac{7}{t - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5t - 15$ $t - 3$ and $t - 3$ The common denominator is $5t - 15$ The denominator of the first term is already $5t - 15$ , so we don't need to change it. To get $5t - 15$ in the denominator of the second ... |
307 | Solve for $x$, $ \dfrac{x - 4}{10x} = \dfrac{4}{5x} - \dfrac{3}{20x} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10x$ $5x$ and $20x$ The common denominator is $20x$ To get $20x$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{x - 4}{10x} \times \dfrac{2}{2} = \dfrac{2x - 8}{20x} $ To get... |
307 | Solve for $y$, $ \dfrac{5}{25y + 25} = -\dfrac{2y - 7}{5y + 5} - \dfrac{1}{20y + 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25y + 25$ $5y + 5$ and $20y + 20$ The common denominator is $100y + 100$ To get $100y + 100$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{5}{25y + 25} \times \dfrac{4}{4} =... |
307 | Solve for $x$, $ -\dfrac{8}{x - 5} = -\dfrac{x - 2}{4x - 20} + \dfrac{10}{x - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $x - 5$ $4x - 20$ and $x - 5$ The common denominator is $4x - 20$ To get $4x - 20$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{8}{x - 5} \times \dfrac{4}{4} = -\dfrac{32}{... |
307 | Solve for $q$, $ \dfrac{4}{q + 2} = \dfrac{q - 7}{5q + 10} + \dfrac{2}{q + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $q + 2$ $5q + 10$ and $q + 2$ The common denominator is $5q + 10$ To get $5q + 10$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{4}{q + 2} \times \dfrac{5}{5} = \dfrac{20}{5q... |
307 | Solve for $r$, $ -\dfrac{6}{r} = -\dfrac{r + 6}{r} - \dfrac{8}{r} $ | If we multiply both sides of the equation by $r$ , we get: $ -6 = -r - 6 - 8$ $ -6 = -r - 14$ $ 8 = -r $ $ r = -8$ |
307 | Solve for $q$, $ -\dfrac{6}{25q - 20} = \dfrac{3q + 2}{10q - 8} + \dfrac{8}{25q - 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25q - 20$ $10q - 8$ and $25q - 20$ The common denominator is $50q - 40$ To get $50q - 40$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{6}{25q - 20} \times \dfrac{2}{2} = -... |
307 | Solve for $a$, $ -\dfrac{6}{2a^3} = \dfrac{2a + 9}{2a^3} + \dfrac{8}{2a^3} $ | If we multiply both sides of the equation by $2a^3$ , we get: $ -6 = 2a + 9 + 8$ $ -6 = 2a + 17$ $ -23 = 2a $ $ a = -\dfrac{23}{2}$ |
307 | Solve for $y$, $ \dfrac{3}{5y} = \dfrac{9}{5y} + \dfrac{y - 9}{5y} $ | If we multiply both sides of the equation by $5y$ , we get: $ 3 = 9 + y - 9$ $ 3 = y$ $ 3 = y $ $ y = 3$ |
307 | Solve for $y$, $ \dfrac{1}{25y} = \dfrac{8}{25y} + \dfrac{y - 6}{15y} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25y$ $25y$ and $15y$ The common denominator is $75y$ To get $75y$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{1}{25y} \times \dfrac{3}{3} = \dfrac{3}{75y} $ To get $75y$ i... |
307 | Solve for $q$, $ -\dfrac{8}{5q - 3} = \dfrac{3q + 6}{20q - 12} + \dfrac{2}{10q - 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5q - 3$ $20q - 12$ and $10q - 6$ The common denominator is $20q - 12$ To get $20q - 12$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{8}{5q - 3} \times \dfrac{4}{4} = -\dfr... |
307 | Solve for $k$, $ -\dfrac{9}{2k - 1} = -\dfrac{9}{2k - 1} - \dfrac{k + 4}{6k - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2k - 1$ $2k - 1$ and $6k - 3$ The common denominator is $6k - 3$ To get $6k - 3$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{9}{2k - 1} \times \dfrac{3}{3} = -\dfrac{27}{... |
307 | Solve for $r$, $ \dfrac{9}{15r + 25} = -\dfrac{r - 5}{12r + 20} - \dfrac{3}{6r + 10} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15r + 25$ $12r + 20$ and $6r + 10$ The common denominator is $60r + 100$ To get $60r + 100$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{9}{15r + 25} \times \dfrac{4}{4} = ... |
307 | Solve for $a$, $ \dfrac{9}{a} = \dfrac{5}{a} - \dfrac{a - 2}{a} $ | If we multiply both sides of the equation by $a$ , we get: $ 9 = 5 - a + 2$ $ 9 = -a + 7$ $ 2 = -a $ $ a = -2$ |
307 | Solve for $q$, $ -\dfrac{4q + 9}{8q} = \dfrac{10}{8q} + \dfrac{8}{2q} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8q$ $8q$ and $2q$ The common denominator is $8q$ The denominator of the first term is already $8q$ , so we don't need to change it. The denominator of the second term is already $8q$ , so we don't need t... |
307 | Solve for $r$, $ \dfrac{3r - 2}{2r + 4} = \dfrac{1}{4r + 8} + \dfrac{10}{2r + 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2r + 4$ $4r + 8$ and $2r + 4$ The common denominator is $4r + 8$ To get $4r + 8$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{3r - 2}{2r + 4} \times \dfrac{2}{2} = \dfrac{6... |
307 | Solve for $t$, $ \dfrac{2}{4t} = \dfrac{3t - 10}{t} - \dfrac{8}{2t} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4t$ $t$ and $2t$ The common denominator is $4t$ The denominator of the first term is already $4t$ , so we don't need to change it. To get $4t$ in the denominator of the second term, multiply it by $\frac... |
307 | Solve for $a$, $ -\dfrac{7}{10a + 10} = \dfrac{5}{2a + 2} + \dfrac{3a - 1}{2a + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10a + 10$ $2a + 2$ and $2a + 2$ The common denominator is $10a + 10$ The denominator of the first term is already $10a + 10$ , so we don't need to change it. To get $10a + 10$ in the denominator of the s... |
307 | Solve for $z$, $ -\dfrac{6}{z - 3} = -\dfrac{z - 6}{z - 3} - \dfrac{10}{2z - 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z - 3$ $z - 3$ and $2z - 6$ The common denominator is $2z - 6$ To get $2z - 6$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{6}{z - 3} \times \dfrac{2}{2} = -\dfrac{12}{2z ... |
307 | Solve for $y$, $ -\dfrac{2y + 10}{y - 2} = -\dfrac{6}{y - 2} - \dfrac{6}{y - 2} $ | If we multiply both sides of the equation by $y - 2$ , we get: $ -2y - 10 = -6 - 6$ $ -2y - 10 = -12$ $ -2y = -2 $ $ y = 1$ |
307 | Solve for $n$, $ \dfrac{4}{n + 5} = -\dfrac{n - 4}{n + 5} - \dfrac{6}{n + 5} $ | If we multiply both sides of the equation by $n + 5$ , we get: $ 4 = -n + 4 - 6$ $ 4 = -n - 2$ $ 6 = -n $ $ n = -6$ |
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