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Solving Equations with Variables on Each Side

After studying this lesson, you will be able to:

  • Solve equations with variables on each side of the equal sign.
  • Solve equations with parentheses and other grouping symbols.

Steps for Solving Equations with Variables on Each Side and with Parentheses

1. Remove parentheses by multiplying

2. Collect like terms on each side of the equal sign

3. Get the variables together on one side of the equation and get the numbers together on the other side of the equation.

4. Isolate the variable by "undoing" the operation (do this until the variable is by itself)

  1. "undo" addition and subtraction first
  2. next, "undo" multiplication and division

5. Check by substituting the solution into the original equation

 

Example 1

3(x + 4) - 5 = 2x - 2 This equation has parentheses, so we have to remove them first and we do so by multiplying (using the distributive property).
3x + 12 - 5 = 2x - 2 After distributing, we no longer have parentheses, but we do have like terms on the left side that need to be collected (12 -5 )
3x + 7 = 2x - 2 Collecting the like terms gives us this equation
3x + 7 - 2x = 2x - 2 - 2x First, we need to get the variables together. It doesn't matter if we put them on the left side or the right side. Let's put them together on the left side this time. To do that, we move 2x to the other side by subtracting 2x from each side. Notice that we line up the like terms. (-2x is lined up with 3x so that it is easier to deal with.)
x + 7 = - 2 After collecting like terms (3x -2x) we now have an equation where the variables are now together. Now, we work this as a 2-step equation.
x + 7 - 7 = - 2 - 7 We need to "undo" +7, so we subtract 7 from each side.
x = -9 This gives us the solution

Check:

substitute -9 for each x in the original equation

3 ( -9 + 4 ) - 5 = 2 (-9 ) -2

3 ( -5 ) -5 = 2 (-9 ) -2 Adding the -9 + 4

-15 - 5 = -18 - 2 Do the multiplication

-20 = -20

 

Example 2

5 + 2(x + 4) = 5(x - 3) + 10 This equation has 2 sets of parentheses, so we have to remove them first and we do so by multiplying (using the distributive property).
5 + 2x + 8 = 5x - 15 + 10 After distributing, we no longer have parentheses, but we do have like terms on the each side that need to be collected (5 + 8 on the left and -15 + 10 on the right )
2x + 13 = 5x - 5 Collecting the like terms gives us this equation
2x + 13 - 2x = 5x - 5 - 2x First, we need to get the variables together. It doesn't matter if we put them on the left side or the right side. Let's put them together on the right side this time. To do that, we move 2x to the other side by subtracting 2x from each side. Notice that we line up the like terms. (-2x is lined up with 5x so that it is easier to deal with.)
13 = 3x - 5 After collecting like terms (5x - 2x) we now have an equation where the variables are now together. Now, we work this as a 2-step equation.
13 + 5 = 3x - 5 + 5 We need to "undo" -5, so we add 5 to each side.

This gives us 18 = 3x

We need to "undo" 3 times x so we divide both sides by 3
6 = x This gives us the solution

Check:

substitute 6 for each x in the original equation

5 + 2 ( 6 + 4 ) = 5 ( 6 - 3 ) + 10

5 + 2 (10) = 5 (3) +10 doing the parentheses first

5 + 20 = 15 + 10 doing the multiplication

25 = 25 doing the addition

 
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