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Step 2.Find the slope of the perpendicular line. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 3.Identify the point. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 4.Substitute the values into the point-slope form,yây1=m(xâx1)yây1=m(xâx1). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 5.Write the equation in slope-intercept form. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
To Graph a Linear InequalityStep 1.Identify and graph the boundary line.If the inequality isâ¤orâ¥â¤orâ¥, the boundary line is solid.If the inequality is < or >, the boundary line is dashed.Step 2.Test a point that is not on the boundary line. Is it a solution of the inequality?Step 3.Shade in one side of the bound... | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 1.Identify and graph the boundary line.If the inequality isâ¤orâ¥â¤orâ¥, the boundary line is solid.If the inequality is < or >, the boundary line is dashed. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 2.Test a point that is not on the boundary line. Is it a solution of the inequality? | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
Step 3.Shade in one side of the boundary line.If the test point is a solution, shade in the side that includes the point.If the test point is not a solution, shade in the opposite side. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-concepts |
boundary line : The line with equationAx+By=CAx+By=Cthat separates the region whereAx+By>CAx+By>Cfrom the region whereAx+By<CAx+By<C. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
geoboard : A geoboard is a board with a grid of pegs on it. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
graph of a linear equation : The graph of a linear equationAx+By=CAx+By=Cis a straight line. Every point on the line is a solution of the equation. Every solution of this equation is a point on this line. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
horizontal line : A horizontal line is the graph of an equation of the formy=by=b. The line passes through they-axis at(0,b)(0,b). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
intercepts of a line : The points where a line crosses thex- axis and they- axis are called the intercepts of the line. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
linear equation : A linear equation is of the formAx+By=CAx+By=C, where A and B are not both zero, is called a linear equation in two variables. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
linear inequality : An inequality that can be written in one of the following forms:Ax+By>CAx+Byâ¥CAx+By<CAx+Byâ¤CAx+By>CAx+Byâ¥CAx+By<CAx+Byâ¤CwhereAandBAandBare not both zero. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
negative slope : A negative slope of a line goes down as you read from left to right. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
ordered pair : An ordered pair(x,y)(x,y)gives the coordinates of a point in a rectangular coordinate system. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
origin : The point(0,0)(0,0)is called the origin. It is the point where thex-axis andy-axis intersect. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
parallel lines : Lines in the same plane that do not intersect. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
perpendicular lines : Lines in the same plane that form a right angle. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
pointâslope form : The pointâslope form of an equation of a line with slopemmand containing the point(x1,y1)(x1,y1)isyây1=m(xâx1)yây1=m(xâx1). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
positive slope : A positive slope of a line goes up as you read from left to right. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
quadrant : Thex-axis and they-axis divide a plane into four regions, called quadrants. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
rectangular coordinate system : A grid system is used in algebra to show a relationship between two variables; also called thexy-plane or the âcoordinate planeâ. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
rise : The rise of a line is its vertical change. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
run : The run of a line is its horizontal change. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
slope formula : The slope of the line between two points(x1,y1)(x1,y1)and(x2,y2)(x2,y2)ism=y2ây1x2âx1m=y2ây1x2âx1. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
slope of a line : The slope of a line ism=riserunm=riserun. The rise measures the vertical change and the run measures the horizontal change. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
slope-intercept form of an equation of a line : The slopeâintercept form of an equation of a line with slopemmandy-intercept,(0,b)(0,b)is,y=mx+by=mx+b. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
solution of a linear inequality : An ordered pair(x,y)(x,y)is a solution to a linear inequality the inequality is true when we substitute the values ofxandy. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
vertical line : A vertical line is the graph of an equation of the formx=ax=a. The line passes through thex-axis at(a,0)(a,0). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
x- intercept : The point(a,0)(a,0)where the line crosses thex- axis; thex- intercept occurs whenyyis zero. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
x-coordinate : The first number in an ordered pair(x,y)(x,y). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
y-coordinate : The second number in an ordered pair(x,y)(x,y). | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
y-intercept : The point(0,b)(0,b)where the line crosses they- axis; they- intercept occurs whenxxis zero. | https://openstax.org/books/elementary-algebra-2e/pages/4-key-terms |
To solve a system of linear equations by graphingStep 1.Graph the first equation.Step 2.Graph the second equation on the same rectangular coordinate system.Step 3.Determine whether the lines intersect, are parallel, or are the same line.Step 4.Identify the solution to the system.If the lines intersect, identify the poi... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 1.Graph the first equation. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 2.Graph the second equation on the same rectangular coordinate system. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 3.Determine whether the lines intersect, are parallel, or are the same line. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 4.Identify the solution to the system.If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.If the lines are parallel, the system has no solution.If the lines are the same, the system has an infinite number of solutions... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 5.Check the solution in both equations. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Determine the number of solutions from the graph of a linear system | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Determine the number of solutions of a linear system by looking at the slopes and intercepts | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Determine the number of solutions and how to classify a system of equations | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Problem Solving Strategy for Systems of Linear EquationsStep 1.Readthe problem. Make sure all the words and ideas are understood.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for. Choose variables to represent those quantities.Step 4.Translateinto a system of equations.Step 5.Solvethe system of ... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 3.Namewhat we are looking for. Choose variables to represent those quantities. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 4.Translateinto a system of equations. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 5.Solvethe system of equations using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Solve a system of equations by substitutionStep 1.Solve one of the equations for either variable.Step 2.Substitute the expression from Step 1 into the other equation.Step 3.Solve the resulting equation.Step 4.Substitute the solution in Step 3 into one of the original equations to find the other variable.Step 5.Write th... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 1.Solve one of the equations for either variable. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 2.Substitute the expression from Step 1 into the other equation. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 3.Solve the resulting equation. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 4.Substitute the solution in Step 3 into one of the original equations to find the other variable. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 5.Write the solution as an ordered pair. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 6.Check that the ordered pair is a solution to both original equations. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
To Solve a System of Equations by EliminationStep 1.Write both equations in standard form. If any coefficients are fractions, clear them.Step 2.Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites.... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 1.Write both equations in standard form. If any coefficients are fractions, clear them. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 2.Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Decide which variable you will eliminate. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Multiply one or both equations so that the coefficients of that variable are opposites. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 3.Add the equations resulting from Step 2 to eliminate one variable. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 4.Solve for the remaining variable. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 5.Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 6.Write the solution as an ordered pair. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 7.Check that the ordered pair is a solution tobothoriginal equations. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Table for coin and mixture applications | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Table for concentration applications | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Table for interest applications | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
To Solve a System of Linear Inequalities by GraphingStep 1.Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true.Step 2.On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality is t... | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 1.Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Graph the boundary line. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Shade in the side of the boundary line where the inequality is true. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 2.On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality is true. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Graph the boundary line. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Shade in the side of that boundary line where the inequality is true. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 3.The solution is the region where the shading overlaps. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
Step 4.Check by choosing a test point. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-concepts |
coincident lines : Coincident lines are lines that have the same slope and samey-intercept. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
complementary angles : Two angles are complementary if the sum of the measures of their angles is 90 degrees. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
consistent system : A consistent system of equations is a system of equations with at least one solution. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
dependent equations : Two equations are dependent if all the solutions of one equation are also solutions of the other equation. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
inconsistent system : An inconsistent system of equations is a system of equations with no solution. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
independent equations : Two equations are independent if they have different solutions. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
solutions of a system of equations : Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x,y). | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
supplementary angles : Two angles are supplementary if the sum of the measures of their angles is 180 degrees. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
system of linear equations : When two or more linear equations are grouped together, they form a system of linear equations. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
system of linear inequalities : Two or more linear inequalities grouped together form a system of linear inequalities. | https://openstax.org/books/elementary-algebra-2e/pages/5-key-terms |
MonomialsA monomial is a term of the formaxmaxm, whereaais a constant andmmis a whole number | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
A monomial is a term of the formaxmaxm, whereaais a constant andmmis a whole number | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
PolynomialspolynomialâA monomial, or two or more monomials combined by addition or subtraction is a polynomial.monomialâA polynomial with exactly one term is called a monomial.binomialâA polynomial with exactly two terms is called a binomial.trinomialâA polynomial with exactly three terms is called a trinomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
polynomialâA monomial, or two or more monomials combined by addition or subtraction is a polynomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
monomialâA polynomial with exactly one term is called a monomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
binomialâA polynomial with exactly two terms is called a binomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
trinomialâA polynomial with exactly three terms is called a trinomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Degree of a PolynomialThedegree of a termis the sum of the exponents of its variables.Thedegree of a constantis 0.Thedegree of a polynomialis the highest degree of all its terms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Thedegree of a termis the sum of the exponents of its variables. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
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