2-1 and 2-2 Solving One- and Two-Step Equations

Quick Review

To solve an equation, get the variable by itself on one side of the equation. You can use properties of equality and inverse operations to isolate the variable. For example, use multiplication to undo its inverse, division.

Example

What is the solution of y 2 + 5 = 8 ? bold italic y over 2 , plus 5 equals 8 question mark

y 2 + 5 − 5 = 8 − 5 Subtract to undo addition . y 2 = 3 Simplify . 2 ⋅ y 2 = 3 ⋅ 2 Multiply to undo division . y = 6 Simplify . table with 4 rows and 3 columns , row1 column 1 , y over 2 , plus 5 minus 5 , column 2 equals 8 minus 5 , column 3 cap subtracttoundoaddition . . , row2 column 1 , y over 2 , column 2 equals 3 , column 3 cap simplify , . , row3 column 1 , 2 dot , y over 2 , column 2 equals 3 dot 2 , column 3 cap multiplytoundodivision . . , row4 column 1 , y , column 2 equals 6 , column 3 cap simplify , . , end table

Exercises

Solve each equation. Check your answer.

6. x + 5 = − 2 x plus 5 equals negative 2
7. a − 2.5 = 4.5 eh minus 2.5 equals 4.5
8. 3b = 42
9. n 5 = 13 n over 5 , equals 13
10. 7 x − 2 = 22.5 7 x minus 2 equals , 22.5
11. y 4 − 3 = − 4 y over 4 , minus 3 equals negative 4
12. 8 + 3 m = − 7 8 plus 3 m equals negative 7
13. − 3 d 4 + 5 = 11 negative , fraction 3 d , over 4 end fraction , plus 5 equals 11
14. Dining Five friends equally split a restaurant bill that comes to $32.50. How much does each pay?
15. Reasoning Justify each step in solving 4 x − 3 = 9 . 4 x minus 3 equals 9 .

    4 x − 3 + 3 = 9 + 3 _ ? 4 x = 12 _ ? 4 x 4 = 12 4 _ ? x = 3 _ ? table with 4 rows and 3 columns , row1 column 1 , 4 x minus 3 plus 3 , column 2 equals 9 plus 3 , column 3 modified under bar with question mark above , row2 column 1 , 4 x , column 2 equals 12 , column 3 modified under bar with question mark above , row3 column 1 , fraction 4 x , over 4 end fraction , column 2 equals , 12 over 4 , column 3 modified under bar with question mark above , row4 column 1 , x , column 2 equals 3 , column 3 modified under bar with question mark above , end table

2-3 Solving Multi-Step Equations

Quick Review

To solve some equations, you may need to combine like terms or use the Distributive Property to clear fractions or decimals.

Example

What is the solution of 12 = 2 x + 4 3 − 2 x 3 ? 12 equals 2 bold italic x plus , 4 thirds , minus , fraction 2 bold italic x , over 3 end fraction , question mark

3 ⋅ 12 = 3 ( 2 x + 4 3 − 2 x 3 ) Multiply by  3 . 36 = 6 x + 4 − 2 x Simplify . 36 = 4 x + 4 Combine like terms . 36 − 4 = 4 x + 4 − 4 Subtract  4 . 32 = 4 x Combine like terms . 32 4 = 4 x 4 Divide each side by  4 . 8 = x Simplify . table with 7 rows and 3 columns , row1 column 1 , 3 dot 12 , column 2 equals 3 . open . 2 x plus , 4 thirds , minus , fraction 2 x , over 3 end fraction . close , column 3 cap multiplyby . 3 . , row2 column 1 , 36 , column 2 equals 6 x plus 4 minus 2 x , column 3 cap simplify , . , row3 column 1 , 36 , column 2 equals 4 x plus 4 , column 3 cap combineliketerms . . , row4 column 1 , 36 minus 4 , column 2 equals 4 x plus 4 minus 4 , column 3 cap subtract . 4 . , row5 column 1 , 32 , column 2 equals 4 x , column 3 cap combineliketerms . . , row6 column 1 , 32 over 4 , column 2 equals , fraction 4 x , over 4 end fraction , column 3 cap divideeachsideby . 4 . , row7 column 1 , 8 , column 2 equals x , column 3 cap simplify , . , end table

Exercises

Solve each equation. Check your answer.

16. 7 ( s − 5 ) = 42 7 open s minus 5 close equals 42
17. 3 a + 2 − 5 a = − 14 3 eh plus 2 minus 5 eh equals negative 14
18. − 4 b − 5 + 2 b = 10 negative 4 b minus 5 plus 2 b equals 10
19. 3.4t + 0.08 = 11
20. 10 = c 3 − 4 + c 6 10 equals , c over 3 , minus 4 plus , c over 6
21. 2 x 7 + 4 5 = 5 fraction 2 x , over 7 end fraction , plus , 4 fifths , equals 5

Write an equation to model each situation. Then solve the equation.

22. Earnings You work for 4 h on Saturday and 8 h on Sunday. You also receive a $50 bonus. You earn $164. How much did you earn per hour?
23. Entertainment Online concert tickets cost $37 each, plus a service charge of $8.50 per ticket. The Web site also charges a transaction fee of $14.99 for the purchase. You paid $242.49. How many tickets did you buy?

Table of Contents