Solving Equations With One Variable

An equation containing one or two variables that represent real numbers is called a linear equation. It is called a linear equation because the equation forms a straight line when graphed.

To solve a linear equation, we need to move all of the terms that contain variables to one side of the equation, and move all of the constants to the other side. Let’s look at an example:

5x+3=2+4x

As you can see above, we have one term on the right side that contains an x. To move this term over to the left side, we’ll subtract 4x from both sides of the equation. This will not change the value of the equation because subtracting the same term from both sides will keep the equation balanced. By subtracting 4x from both sides of the equation, the 4x on the right side of the equation will cancel out:

5x+3-4x=2+4x-4x

Simplify the right side of the equation:

5x+3-4x=2+4x-4x
5x+3-4x=2+0
5x+3-4x=2

Subtracting 4x from 4x will give us 0, and remove the variable from the right side of the equation. Now let’s simplify the left side of the equation as well:

5x+3-4x=2

Since the 5x and -4x are like terms, they can be combined to give us x:

x+3=2

With the left side of the equation containing the variable, we now need to move all of the constants to the right side of the equation. We’ll use the same technique as above to remove the 3 from the left side:

x+3=2
x+3-3=2-3

In this case, we subtracted 3 from both sides of the equation in order to cancel out the 3 on the left side.

x=2-3

Finally, let’s simplify the right side of the equation:

x=-1

And we’re finished! The solution to this equation is x=-1

Solving Equations With Multiple Variables

Now that we have learned how to solve equations that have one variable, let’s try solving an equation with two variables (since a linear equation is commonly written in the form y=mx+b, we’ll use the variables y and x):

4y+3=9x-3+2y

Just as before, we want to get all of the variables on the left hand side. But now that we have two variables, we need to choose which one we want to solve for. When dealing with linear equations, we usually want to solve for y. Since we are solving for y, we need to move all of the terms that contain y to the left side of the equation:

Subtract 2y from both sides of the equation:
4y-2y+3=9x-3+2y-2y

Simplify the left side of the equation by subtracting 2y from 4y:
2y+3=9x-3+2y-2y

Simplify the right side of the equation by subtracting 2y from 2y:
2y+3=9x-3

Now we have all the y terms on the left side of the equation, and all of the x terms on the right side of the equation. Next we will move all of the constants to the right side of the equation to isolate the y term:

Subtract 3 from both sides of the equation:
2y+3-3=9x-3-3

Then simplify:
2y=9x-3-3
2y=9x-6

We’re almost done - the y term still has a coefficient of 2. To isolate the y, we need to remove the 2 from the y term. To do this, we will divide each term in the equation by 2:

2y/2=9x/2-6/2

Then simplify:

y=9x/2-3

And we’re done! The solution to the equation 4y+3=9x-3+2y when solving for y is y=9x/2-3.