How do you solve equations with variables on both sides

Solving algebra equations with variables on both sides is really tough. But, you will have no problems solving these equations after you follow the step-by-step instructions in these 3 examples.

So far, you've learned how to solve one-step equations, two-step equations, and equations with fractions. If you haven't mastered these skills or need a refresher, please go back and review these lessons before attempting this one! It will be so much easier if you have the background!

So, if you are ready, let's move on!

Solving algebra equations with variables on both sides look intimidating, but they really aren't that hard if you have the background skills. In fact, I think these types of equations are fun because it's like moving pieces of a puzzle around.

This is what you need to remember:

Solving Equations With Variables on Both Sides

• If you have any fractions, get rid of those first by multiplying ALL terms by the denominator.
• Use the distributive property if needed.
• Your ultimate goal is to get all of the constants on one side of the equation and all of the variables on the other side of the equation. You can accomplish this by adding or subtracting terms on BOTH sides of the equation.

Let's take a look at a few examples.

Example 1 - Variables on Both Sides

Did you notice how we used all of the rules we learned in previous lessons?

Remember, our goal is to get all of the variables on one side and all the constants on the other side.

This next example shows how to solve if the distributive property is involved.

Example 2 - Working with the Distributive Property

Yes, this is a confusing topic, so if you are still having trouble, check out example 2 on video.

Now let's see what happens when we bring it all together and try to solve an equation that has fractions and variables on both sides!

Example 3 - Fractions and Variables on Both Sides

It doesn't look too hard, does it? Remember: it's like moving puzzle pieces around - all the variable terms on one side and all the constant terms on the other side!

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Solving Equations with a Variable on Both Sides

Sometimes, the unknown quantity will appear on both sides of an equation. This is where the properties learned in 5.1 and 5.2 come in handy. A quantity with a variable can be treated just like a quantity without variables -- a quantity with a variable follows all the rules learned in the last two sections. For example, we can add a quantity with a variable to both sides without changing the equation or the values that make it true:

15 - x = 4x
15 - x + x = 4x + x
15 + 0x = 5x
15 = 5x
3 = x
x = 3

Similarly, we can subtract a term with a variable from both sides of the equation:

5x = 6 + 2x
5x - 2x = 6 + 2x - 2x
3x = 6 + 0x
3x = 6
x = 2

After simplifying, the first step in solving an equation with a variable on both sides is to get the variable on one side. This is done by reversing the addition or subtraction of one of the terms with the variable. In other words, we must add to both sides or subtract from both sides one of the quantities that contains the variable. It is generally easier to add or to subtract the smaller quantity from the larger quantity, so we are working with positive coefficients, but either way works. Once the variable is on one side only, we can proceed using inverse operations, as in 4.1 and 4.2.

Example 1. Solve for x: 3x + 2x = 12 - x

• Simplify: 5x = 12 - x
• Get the variable on one side:
  • 5x + x = 12 - x + x
  • 6x = 12
• Solve using inverse operations:
• Check: 3(2) + 2(2) = 12 - 2 ? Yes!

Example 2. Solve for y: 5y - 3 = 3y + 5

• The equation is already simplified.
• Get the variable on one side:
  • 5y - 3 - 3y = 3y + 5 - 3y
  • 5y - 3y - 3 = 3y - 3y + 5
  • 2y - 3 = 5
• Solve using inverse operations:
• Check: 5(4) - 3 = 3(4) + 5 ? Yes!