Solving Linear Equations

July 2003

Mathematics Bridge Program Algebra

Solving Linear Equations

1. Definitions

Linear Equation Solution Property of Equality

2. Solving Linear Equations

Distributive Property Eliminating Fractions

3. Solving for One Variable in a Formula

4. Summary: Process for Solving Linear Equations

5. Worked out Solutions for Exercises

Page 1 of 14

1

Definitions:

Linear Equations: An equation is a statement that two expressions have the same value:

expression on the left = expression on the right Example Example x + 7 = 2x − 3 15 = 7 + 8

Any number or the value of the variable that makes an equation a true statement is called a solution of the equation. The process of finding the solution of an equation is called solving the equation for the variable or unknown. You can also call it finding x (or y, or whatever variable you are using). All solutions should be checked by substituting back into the original equation, and seeing if that will give a true statement. If you solved x + 5 = 9 to give you x = 2 , you will easily see if you have the right answer by substituting x = 2 back into x + 5 = 9 (you don’t).

Linear equation in one variable (with one unknown): A linear equation in one variable can be written in the form

ax + b = c ,

where a, b, and c are real numbers and a ≠ 0.

Note that the variable (x) is raised to the first power: that’s how you recognize a linear equation. Examples of linear equations: 3x + 5 = 9

x + 3x + 1 can be manipulated to be 4 written in the standard form of ax + b = c 2(x − 5) =

Property of Equality: If a, b, and c are real numbers,

a = b and a + c = b + c are equivalent equations a = b and ac = bc are equivalent equations

a b = are equivalent equations, c ≠ 0 c c This property guarantees that by adding, subtracting (as in adding a negative), multiplying or dividing by the same quantity on both sides you end up with equivalent equations and the solution of the equation is not changed. a...