Math CentralQuandaries & Queries


Question from Allen, a student:

I am having a problem solving for x. The problem is: 9x + 7 = 5x - 3

can someone please help?

Hi Allen.

The goal is to have x by itself (isolated) on one side of the equal sign and just a number on the other side.

When you work with equations, the most important rule to remember is this:

Whatever you do to one side of the equation, you must do to the other side as well.

The first step is to get rid of the 5x on the right side. To do that mathematically, we have to subtract it from the left side, because 5x - 5x = 0, so it goes away. But if we subtract 5x from the right side, we must subtract it also from the left side:

9x + 7 = 5x - 3

9x + 7 - 5x = 5x - 3 - 5x

You can only add and subtract terms that have the same variables (letters) with them. So 9x cannot be added to 7 to make 16x, but 9x - 5x is 4x. So the line above simplifies to:

4x + 7 = -3

Now we must get rid of the + 7 on the right side. We can subtract 7 to do that (do it on both sides):

4x + 7 - 7 = -3 - 7

This simplifies to

4x = -10

We want finally to get rid of the 4, but that's not added or subtracted, so we can't just add or subtract 4 from both sides. It is multiplied, so to get rid of it, we can divide both sides by 4. That way the fours cancel on the right hand side:

4x ÷ 4 = -10 ÷ 4

x = -2.5.

That's how you solve these kinds of problems, Allen.

Stephen La Rocque. >

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