Solving and Writing Linear Equations

To solve a linear equation, use the properties of equality and properties of real numbers to find the value of the variable that satisfies the equation.

Example 1

Algebra Solve each equation.

1. 5 x − 3 = 2 5 bold italic x minus 3 equals 2

   5 x − 3 = 2   5 x = 5 Add  3  to each side . x = 1 Divide each side by  5 . table with 3 rows and 4 columns , row1 column 1 , 5 x minus 3 , column 2 equals , column 3 2 , column 4 , row2 column 1 , 5 x , column 2 equals , column 3 5 , column 4 cap add , 3 . toeachside . . , row3 column 1 , x , column 2 equals , column 3 1 , column 4 cap divideeachsideby . 5 . , end table

2. 1 − 2 ( x + 1 ) = x 1 minus 2 open bold italic x plus 1 close equals bold italic x

   1 − 2 ( x + 1 ) = x   1 − 2 x − 2 = x Use the Distributive Property . − 1 − 2 x = x Simplify the left side . − 1 = 3 x Add  2 x  to each side . − 1 3 = x Divide each side by  3 . table with 5 rows and 4 columns , row1 column 1 , 1 minus 2 open x plus 1 close , column 2 equals , column 3 x , column 4 , row2 column 1 , 1 minus 2 x minus 2 , column 2 equals , column 3 x , column 4 cap usethecap distributivecap property . . , row3 column 1 , negative 1 minus 2 x , column 2 equals , column 3 x , column 4 cap simplifytheleftside . . , row4 column 1 , negative 1 , column 2 equals , column 3 3 x , column 4 cap add , 2 x . toeachside . . , row5 column 1 , negative , 1 third , column 2 equals , column 3 x , column 4 cap divideeachsideby . 3 . , end table

You will sometimes need to translate word problems into equations. Look for words that suggest a relationship or some type of mathematical operation.

Example 2

Algebra A student has grades of 80, 65, 78, and 92 on four tests. What is the minimum grade she must earn on her next test to ensure an average of 80?

The student must earn 85 on the next test for an average of 80.

Exercises

Algebra Solve each equation.

1. 3 n + 2 = 17 3 n plus . 2 equals 17
2. 5 a − 2 = − 12 5 eh minus . 2 equals , negative 12
3. 2 x + 4 = 10 2 x plus . 4 equals 10
4. 3 ( n − 4 ) = 15 3 open n minus 4 . close , equals 15
5. 4 + 2 y = 8 y 4 plus . 2 y equals 8 y
6. − 6 z + 1 = 13 − 3 z negative 6 z plus . 1 equals . 13 minus 3 z
7. 6 − ( 3 t + 4 ) = t 6 minus . open 3 t plus 4 . close , equals t
8. 7 = − 2 ( 4 n − 4.5 ) 7 equals , minus . 2 open 4 n minus 4.5 close
9. ( w + 5 ) − 5 = ( 2 w + 5 ) open w plus 5 , close minus , 5 equals . open 2 w plus 5 close
10. 5 7 p − 10 = 30 5 sevenths , p minus . 10 equals 30
11. m − 3 − 3 = 1 m over negative 3 , minus 3 equals 1
12. 5 k + 2 ( k + 1 ) = 23 5 k plus . 2 open k plus 1 . close , equals 23
13. Twice a number subtracted from 35 is 9. What is the number?
14. The Johnsons pay $ 9.95 dollars , 9.95  a month plus $ .035 dollars , .035  per min for local phone service. Last month, they paid $ 12.75 . dollars , 12.75 , .  How many minutes of local calls did they make?

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