Linear Equations

Linear Equations

The overriding principle in solving linear equations is to isolate the variable.

At times, the variable expressions or other like terms may need to be combined first.   Then the variable can be isolated by performing inverse operations to both sides of the equation.

When solving linear equations, clearly document each step in the process so that you can avoid mistakes or catch errors when checking your work.

Example 1

Solve:     13 − (2x + 2) = 2(x + 2) + 3x

Multiply across the parentheses using the distributive property.  If there are minus signs in front of parentheses, they get distributed as well:

13 − (2x + 2) = 2(x + 2) + 3x

13 − 2x − 2 = 2x + 4 + 3x

Combine like terms:

13 − 2x 2 = 2x + 4 + 3x

11 − 2x = 5x + 4

Get the variables on one side of the equation, and the constants (terms without variables) on the other side by performing inverse operations on both sides:

11 − 2x + 2x = 5x + 4 + 2x

11 = 7x + 4

11 4 = 7x + 4 4

7 = 7x

Perform one more inverse operation (either multiplication or division) to get the variable by itself:

7 ÷ 7 = 7x ÷ 7

1 = x

Example 2

Solve:    -2x − 5 = -3

First, add 5 to both sides:
-2x − 5 + 5 = -3 + 5

Then multiply both sides by :
-2x  () = 2  ()

x = -1

Example 3

Solve:    (x + 1) = 1

Multiply both sides by the LCD, 5.

• 5 = 1 • 5

x + 1 = 5

x = 4

If the equation contains a variable on both sides, add or subtract the variable from one of the sides to eliminate it.

Example 4

Practice

Solve the following equations.

1) 5n − 2(n + 4) = 3

2) 16 = 4x + 6 − 2x

3) x + 1.24 − 0.07x = 4.96

4)

5) 1 + (3 + x) + 6x = 6

6) 3(x − 2) − 4(2x + 3) = 2

7) (t − 5) = -6

8)  2(n + 3) = -3n + 10

9)  2(x − 5) + 3 = 3x + 9

10)  5.74x + 5.42 = 2.24x − 9.28

11)

12)

13)  (2x + 3) = (x − 6) + 4

14)  12 − 6w + 3(2w + 3) = 2w + 5

15)

1.  n =

2.  x = 5

3.  x = 4

4.  x =

5.  x =

6.  x = −4

7.  t = −13

8.  n =

9.  x = −16

10.  x = −4.2

11.  k =

12.  x = 4

13.  x =

14.  w = 8

15.  n = 0