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


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  ()

x1

Example 3 - Solve for the following expression

Multiply both sides by the LCD, 5.

x + 1 = 5

x = 4



Example 4