General Steps in Solving a Linear Equation

1. Simplify each side of the equation by removing grouping symbols and combining like terms.

2. Isolate variable terms on one side of the equation and constant terms on the other side. To accomplish this, you may add (or subtract) the same real number or variable expression on both sides of the equation.

3. Isolate the variable. To accomplish this, you may multiply (or divide) on both sides of the equation by the same nonzero quantity.

4. Check.

Example. Solve: a) 4(2x + 1) – 29 = 3(2x – 5)

b) 5(x – 9) = 5x – 45

c) 3x + 6 – (3x – 2) = 5 + 4x + 4(3 – x)

d)

Solution: a) 4(2x + 1) – 29 = 3(2x – 5)

8x + 4 – 29 = 6x – 15 Remove parentheses, using distributive property

8x – 25 = 6x – 15 Combine like terms

8x – 25 – 6x = 6x – 15 – 6x Subtract 6x from both sides; this will

isolate variable terms

2x – 25 = -15 Combine like terms

2x – 25 + 25 = -15 + 25 Add 25 to both sides; this will isolate

constant terms

2x = 10

Divide both sides by 2 to isolate the

variable

x = 5

Check: 4(2x + 1) – 29 = 3(2x – 5) Original Equation

4(2·5 + 1) – 29 = 3(2·5 – 5) The solution, 5, is substituted for x

4(10 + 1) – 29 = 3(10 – 5) Each side of the equation is simplified

4(11) – 29 = 3(5)

44 – 29 = 15

15 = 15 This true statement indicates that 5 is the correct solution.

b) 5(x – 9) = 5x – 45

5x – 45 = 5x – 45

Since the left side of the equation is identical to the right side of the equation, the given equation is true for every value of x. This equation is an identity. The solution set consists of all real numbers.

c) 3x + 6 – (3x – 2) = 5 + 4x + 4(3 – x)

3x + 6 – 3x + 2 = 5 + 4x + 12 – 4x

8 = 17

This is a false statement. So this equation is a contradiction. There is no solution.

d) When the equation contains fractions, multiply

both sides by the LCD, 12m

6m - 12 = m(6m + 5)

6m - 12 = 6m + 5m

6m - 12 - 6m = 6m + 5m - 6m

-12 = 5m