PDF Solving Multi-Step Equations With Variables on Both Sides ...

Solving Multi-Step Equations With Variables on Both Sides When solving multi-step equations you need to: 1) Eliminate parenthesis by applying the Distributive Property 2) Simplify both sides of the equation by Combining Like Terms. In other words, combine

similar variables and constants together.

3) Decide where you want to keep the variable(left or right side); that helps you decide

where to keep the constants (opposite side where the variable is located).

4) Cancel out numbers by applying inverse (opposite) operations: addition and subtraction

are opposite operations as in the case of multiplication and division. ***Remember when you cross the line, CHANGE the sign!!!!

Example 1: Solve the multi-step equation

Solution: This is a typical problem in multi-step equations where there are variables on both sides. Notice that there are no parenthesis in this equation and nothing to combine like terms in either both sides of the equation. Clearly, our first step is to decide where to keep or isolate the unknown variable x. Since 7x is "larger" than 2x, then we might as well keep it on the left side.

This means we have to get rid of the 2x on the right side. To do that, we need to subtract both sides by 2x because the opposite of +2x is -2x.

After simplifying by subtracting both sides by 2x, we have...

It's nice to see just the variable x on the left side. This implies that we have to move all the constants to the right side and that +3 on the left must be removed. The opposite of +3 is -3, therefore, we will subtract both sides by 3.

After subtracting both sides by 3, we get...

The last step is to isolate variable x by itself on the left side of the equation. Since +5 is multiplying x, then its opposite operation is to divide by +5. So, we are going to divide both sides by 5 and then we are done!

Example 2: Solve the multi-step equation

Solution: Our first step should be to get rid of the parenthesis by applying the Distributive property. That is, multiply -2 inside each term of the parenthesis (-4x + 5).

Now, it is time to decide where to keep the unknown variable x. If you decide to keep the variable on the left side, that is perfectly fine. However for practice, let's try keeping it on the right side. We should arrive at the same answers.

To get rid of the -3x on the left side, we add both sides by 3x since the opposite of -3x is +3x. After we simplify by adding both sides by 3x, we obtain this less messy equation. It's nice to see the variable x just on the right side. So, we have to move all the constants to the left side. Clearly, the -10 on the right must be removed. The opposite of -10 is +10, therefore, we will add both sides by 10. After adding both sides by 10, the linear equation becomes...

The last step is to isolate variable x by itself on the right side of the equation. Since +11 is multiplying x, then its opposite operation is to divide by +11. So, we are going to divide both sides by 11 and we are done!

Example 3: Solve the multi-step equation

Solution:

Our first step should be to eliminate the parentheses on BOTH sides of the equation by applying the Distributive property. For the left side, multiply -4 inside each term of the parenthesis (-8 + 4x) and for the right side, multiply -3 inside the parenthesis (1 + 8x).

Now, before we even decide which side of the equation to isolate the variable, it looks like we have to perform some house cleaning. We need to combine like terms (the x's) on the left side of the equation. Again it doesn't matter which side to isolate the variable being solved. Say, we decided to keep it on the left.

That means we have to get rid of the -24x on the right side. The opposite of -24x is +24x so we are going to add both sides by 24x.

Next, we have to move all the constants to the right side of the equation. That +32 on the left side must go! The opposite of +32 is -32, so then we will subtract both sides by 32.

The last step is to isolate variable x by itself on the left side of the equation. Since +5 is multiplying x, then its opposite operation is to divide by +5. And so, let's divide both sides by 5 and we are done!

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