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Step 6.Check the answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 7.Answer the question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
The Division Property of EqualityâFor any numbersa,b, andc, andcâ0câ0, ifa=ba=b, thenac=bcac=bc.When you divide both sides of an equation by any non-zero number, you still have equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
The Multiplication Property of EqualityâFor any numbersa,b, andc, ifa=ba=b, thenac=bcac=bc.If you multiply both sides of an equation by the same number, you still have equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the EquationStep 1.Choose which side will be the âvariableâ sideâthe other side will be the âconstantâ side.Step 2.Collect the variable terms to the âvariableâ side of the equation, using the Addition or Subtraction P... | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 1.Choose which side will be the âvariableâ sideâthe other side will be the âconstantâ side. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 2.Collect the variable terms to the âvariableâ side of the equation, using the Addition or Subtraction Property of Equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 3.Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 4.Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 5.Check the solution by substituting it into the original equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
General Strategy for Solving Linear EquationsStep 1.Simplify each side of the equation as much as possible.Use the Distributive Property to remove any parentheses.Combine like terms.Step 2.Collect all the variable terms on one side of the equation.Use the Addition or Subtraction Property of Equality.Step 3.Collect all ... | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 1.Simplify each side of the equation as much as possible.Use the Distributive Property to remove any parentheses.Combine like terms. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 2.Collect all the variable terms on one side of the equation.Use the Addition or Subtraction Property of Equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 3.Collect all the constant terms on the other side of the equation.Use the Addition or Subtraction Property of Equality. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 4.Make the coefficient of the variable term to equal to 1.Use the Multiplication or Division Property of Equality.State the solution to the equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 5.Check the solution.Substitute the solution into the original equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Strategy to Solve an Equation with Fraction CoefficientsStep 1.Find the least common denominator of all the fractions in the equation.Step 2.Multiply both sides of the equation by that LCD. This clears the fractions.Step 3.Solve using the General Strategy for Solving Linear Equations. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 1.Find the least common denominator of all the fractions in the equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 2.Multiply both sides of the equation by that LCD. This clears the fractions. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 3.Solve using the General Strategy for Solving Linear Equations. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
To Solve an Application (with a formula)Step 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 a variable to represent that quantity.Step 4.Translateinto an equation. Write the appropriate formula for the situation. Subs... | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 4.Translateinto an equation. Write the appropriate formula for the situation. Substitute in the given information. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Distance, Rate and TimeFor an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:d=rtd=rtwhered= distance,r= rate,t= time. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
To solve a formula for a specific variablemeans to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Subtraction Property of InequalityFor any numbers a, b, and c,ifa<ba<bthenaâc<bâcaâc<bâcandifa>ba>bthenaâc>bâc.aâc>bâc. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Addition Property of InequalityFor any numbers a, b, and c,ifa<ba<bthena+c<b+ca+c<b+candifa>ba>bthena+c>b+c.a+c>b+c. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
Division and Multiplication Properties of InequalityFor any numbers a, b, and c,ifa<ba<bandc>0c>0, thenac<bcac<bcandac>bcac>bc.ifa>ba>bandc>0c>0, thenac>bcac>bcandac>bcac>bc.ifa<ba<bandc<0c<0, thenac>bcac>bcandac>bcac>bc.ifa>ba>bandc<0c<0, thenac<bcac<bcandac<bcac<bc. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
When wedivide or multiplyan inequality by a:positivenumber, the inequality stays thesame.negativenumber, the inequalityreverses. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
positivenumber, the inequality stays thesame. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
negativenumber, the inequalityreverses. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-concepts |
conditional equation : An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-terms |
contradiction : An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-terms |
identity : An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-terms |
solution of an equation : A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. | https://openstax.org/books/elementary-algebra-2e/pages/2-key-terms |
Problem-Solving StrategyStep 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 a variable to represent that quantity.Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the... | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Consecutive IntegersConsecutive integers are integers that immediately follow each other.n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.Consecutive even integers are even integers that immediately follow one... | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Percent IncreaseTo find the percent increase:Step 1.Find the amount of increase.increase=new amountâoriginalamountincrease=new amountâoriginalamountStep 2.Find the percent increase. Increase is what percent of the original amount? | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Find the amount of increase.increase=new amountâoriginalamountincrease=new amountâoriginalamount | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Find the percent increase. Increase is what percent of the original amount? | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Percent DecreaseTo find the percent decrease:Step 1.Find the amount of decrease.decrease=original amountânewamountdecrease=original amountânewamountStep 2.Find the percent decrease. Decrease is what percent of the original amount? | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Find the amount of decrease.decrease=original amountânewamountdecrease=original amountânewamount | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Find the percent decrease. Decrease is what percent of the original amount? | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Simple InterestIf an amount of money,P, called the principal, is invested for a period oftyears at an annual interest rater, the amount of interest,I,earned isI=PrtwhereI=interestP=principalr=ratet=timeI=PrtwhereI=interestP=principalr=ratet=time | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Discountamount of discount is discount rate··original pricesale price is original price â discount | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
amount of discount is discount rate··original price | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
sale price is original price â discount | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Mark-upamount of mark-up is mark-up rate··original costlist price is original cost + mark up | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
amount of mark-up is mark-up rate··original cost | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
list price is original cost + mark up | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the model.number·value=totalvaluenumber·value=totalvaluewherenumberis the number of coins andvalueis the value of each coin;total valueis the total value of all the coins | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Problem-Solving StrategyâCoin Word ProblemsStep 1.Readthe problem. Make all the words and ideas are understood. Determine the types of coins involved.Create a table to organize the information.Label the columns type, number, value, total value.List the types of coins.Write in the value of each type of coin.Write in t... | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Readthe problem. Make all the words and ideas are understood. Determine the types of coins involved.Create a table to organize the information.Label the columns type, number, value, total value.List the types of coins.Write in the value of each type of coin.Write in the total value of all the coins. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Create a table to organize the information. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Label the columns type, number, value, total value. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
List the types of coins. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Write in the value of each type of coin. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Write in the total value of all the coins. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table.Multiply the number times the value to get the total value of each type of coin. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.Write the equation by adding the total values of all the types of coins. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Problem-Solving Strategy for Geometry ApplicationsStep 1.Readthe problem and make all the words and ideas are understood. Draw the figure and label it with the given information.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for by choosing a variable to represent it.Step 4.Translateinto an equat... | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Readthe problem and make all the words and ideas are understood. Draw the figure and label it with the given information. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 3.Namewhat we are looking for by choosing a variable to represent it. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 4.Translateinto an equation by writing the appropriate formula or model for the situation. Substitute in the given information. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Triangle Properties Forâ³ABCâ³ABCAngle measures:mâA+mâB+mâC=180mâA+mâB+mâC=180Perimeter:P=a+b+cP=a+b+cArea:A=12bh,b=base,h=heightA=12bh,b=base,h=heightA right triangle has one90°90°angle. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
mâA+mâB+mâC=180mâA+mâB+mâC=180 | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
P=a+b+cP=a+b+c | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
A=12bh,b=base,h=heightA=12bh,b=base,h=height | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
The Pythagorean TheoremIn any right triangle,a2+b2=c2a2+b2=c2wherecis the length of the hypotenuse andaandbare the lengths of the legs. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Properties of RectanglesRectangles have four sides and four right (90°) angles.The lengths of opposite sides are equal.The perimeter of a rectangle is the sum of twice the length and twice the width:P=2L+2W.P=2L+2W.The area of a rectangle is the length times the width:A=LW.A=LW. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Rectangles have four sides and four right (90°) angles. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
The lengths of opposite sides are equal. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
The perimeter of a rectangle is the sum of twice the length and twice the width:P=2L+2W.P=2L+2W.The area of a rectangle is the length times the width:A=LW.A=LW. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Distance, Rate, and TimeD=rtwhereD= distance,r= rate,t= time | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
D=rtwhereD= distance,r= rate,t= time | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Problem-Solving StrategyâDistance, Rate, and Time ApplicationsStep 1.Readthe problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you... | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you know. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity.Complete the chart.Use variable expressions to represent that quantity in each row.Multiply the rate times the time to get the distance. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
Step 4.Translateinto an equation.Restate the problem in one sentence with all the important information.Then, translate the sentence into an equation. | https://openstax.org/books/elementary-algebra-2e/pages/3-key-concepts |
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