# Solving Linear Equations in One Variable

**5 Combining Like Terms and Simplifying**

Linear equations typically will not be given in standard form and thus will
require some additional preliminary

steps. These additional steps are to first simplify the expressions on each side
of the equal sign using the

order of operations.

**5.1 Opposite Side Like Terms**

Given a linear equation in the form ax + b = cx + d we must combine like
terms on opposite sides of the

equal sign. To do this we will use the addition or subtraction property of
equality to combine like terms on

either side of the equation.

**Example 10
**Solve for y: -2y + 3 = 5y + 17

-2y + 3 = 5y + 17 | |

-2y + 3 - 5y = 5y + 17 - 5y | Subtract 5y on both sides. |

-7y + 3 = 17 | |

-7y + 3 - 3 = 17 - 3 | Subtract 3 on both sides. |

-7y = 14 | |

Divide both sides by -7. | |

y = -2 |

The solution set is
.

**5.2 Same Side Like Terms
**

We will often encounter linear equations where the expressions on each side of the equal sign could be

simplified. If this is the case then it is usually best to simplify each side first. After which we then use the

properties of equality to combine opposite side like terms.

**Example 11**

Solve for a:-4a + 2 - a = 3 + 5a - 2

-4a + 2 - a = 3 + 5a - 2 | Add same side like terms first. |

-5a + 2 = 5a + 1 | |

-5a + 2 - 5a = 5a + 1 - 5a | Subtract 5a from both sides. |

-10a + 2 = 1 | |

-10a + 2 - 2 = 1 - 2 | Subtract 2 from both sides. |

-10a = -1 | |

Divide both sides by -10. | |

The solution set is
.

**5.3 Simplifying Expressions First
**

When solving linear equations the goal is to determine what value, if any, will solve the equation. A general

guideline is to use the order of operations to simplify the expressions on both sides first.

**Example 12**

Solve for x: 5 (3x + 2) - 2 = -2 (1 - 7x)

5 (3x + 2) - 2 = -2 (1 - 7x) | Distribute. |

5 (3x + 2) - 2 = -2 (1 - 7x) | Add same side like terms. |

15x + 10 - 2 = -2 + 14x | |

15x + 8 = -2 + 14x | |

15x + 8 - 14x = -2 + 14x - 14x | Subtract 14x on both sides. |

x + 8 = -2 | |

x + 8 - 8 = -2 - 8 | Subtract 8 on both sides. |

x = -10 |

The solution set is .

**Figure 2:** Video Example 02

**6 Conditional Equations, Identities, and Contradictions
**

There are three different kinds of equations defined as follows.

**Definition 3: Conditional Equation**

A conditional equation is true for particular values of the variable.

**Definition 4: Identity**

An identity is an equation that is true for all possible values of the variable. For example, x = x

has a solution set consisting of all real numbers, .

**Definition 5:**Contradiction

A contradiction is an equation that is never true and thus has no solutions. For example, x + 1 =

x has no solution. No solution can be expressed as the empty set f

So far we have seen only conditional linear equations which had one value in the solution set. If when

solving an equation and the end result is an identity, like say 0 = 0, then any value will solve the equation.

If when solving an equation the end result is a contradiction, like say 0 = 1, then there is no solution.

**Example 13**

Solve for x:4 (x + 5) + 6 = 2 (2x + 3)

4 (x + 5) + 6 = 2 (2x + 3) | Distribute |

4x + 20 + 6 = 4x + 6 | Add same side like terms. |

4x + 26 = 4x + 6 | |

4x + 26 - 4x = 4x + 6 - 4x | Subtract 4x on both sides. |

26 = 6 | False |

There is no solution,
.

**Example 14**

Solve for y:3 (3y + 5) + 5 = 10 (y + 2) - y

3 (3y + 5) + 5 = 10 (y + 2) - y | Distribute |

9y + 15 + 5 = 10y + 20 - y | Add same side like terms. |

9y + 20 = 9y + 20 | |

9y + 20 - 20 = 9y + 20 - 20 | Subtract 20 on both sides. |

9y = 9y | |

9y - 9y = 9y - 9y | Subtract 9y on both sides. |

0 = 0 | True |

The equation is an identity, the solution set consists of
all real numbers, .

**7 Linear Literal Equations**

Literal equations, or formulas, usually have more than one variable. Since the
letters are placeholders for

values, the steps for solving them are the same. Use the properties of equality
to isolate the indicated

variable.

**Example 15**

Solve for a: P = 2a + b

P = 2a + b | |

P - b = 2a + b - b | Subtract b on both sides. |

P - b = 2a | |

Divide both sides by 2. | |

Solution:

**Example 16**

Solve for x:

Multiply both sides by 2. | |

2z = x + y | |

2z - y = x + y - y | Subtract y on both sides. |

2z - y = x |

Solution x = 2z - y

**8 Exercises
8.1 Checking Solutions**

Exercise 1Is x = 7 a solution to - 3x + 5 = -16? |
(Solution on p. 12.) |

Exercise 2Is x = 2 a solution to - 2x - 7 = 28? |
(Solution on p. 12.) |

Exercise 3Is x = -3 a solution to |
(Solution on p. 12.) |

Exercise 4Is x = -2 a solution to 3x - 5 = -2x - 15? |
(Solution on p. 12.) |

Exercise 5Is x = -1/2 a solution to 3 (2x + 1) = -4x - 3? |
(Solution on p. 12.) |

**8.2 Solving in One Step**

Exercise 6Solve for x: x - 5 = -8 |
(Solution on p. 12.) |

Exercise 7Solve for y: - 4 + y = -9 |
(Solution on p. 12.) |

Exercise 8Solve for x: |
(Solution on p. 12.) |

Exercise 9Solve for x: |
(Solution on p. 12.) |

Exercise 10Solve for x: 4x = -44 |
(Solution on p. 12.) |

Exercise 11Solve for a: - 3a = -30 |
(Solution on p. 12.) |

Exercise 12Solve for y: 27 = 9y |
(Solution on p. 12.) |

Exercise 13Solve for x: |
(Solution on p. 12.) |

Exercise 14Solve for t: |
(Solution on p. 12.) |

Exercise 15Solve for x: |
(Solution on p. 12.) |