Solving an equation

An equation is in simplified form when there is no multiplication, division, addition, or subtraction left to perform on either side of the equation and the variable is on one side only. For example, 4(3x+2) = 15 is not in simple form until we multiply four through the parentheses. And, the equation 3x +2 = 9(x – 2) is not in simple form because the variable is on both sides of the equation and the nine must be multiplied through the parentheses.

Here are some equations which are in simple form:

[math]3x + 5 = 12[/math]

[math]-2x + 8 = 12[/math]

[math]28 = -4x + 4[/math]

Note that there is no multiplication and division to perform and there are no like terms to combine (addition or subtraction) on either side of the equation. So, we can proceed to solve the equation, which will involve some operations.

To solve an equation with both sides in simplified form, we are essentially working backward to undo the operations that were performed on the variable. In working backward, our order of operations—Please Excuse My Dear Aunt Sally—are performed in reverse order. So, addition and subtraction are performed before multiplication and division.

Example 1:    [math]3x + 1 = 7[/math]

Look on the side of the equation with the variable. The variable was multiplied by three and then one was added. Let’s undo that by performing the opposite operations, working backwards. That is, we subtract one from both sides of the equation and then divide both sides by three.

Step 1: We start with the subtraction. (Instead of thinking of subtraction, we could say that we are combining the +1 with a -1.)

[math]\begin{align} 3x + 1 &= 7 \\ 3x +1 -1 &= 7 -1 \\ 3x &= 6 \\ \end{align}[/math]

Step 2: Three is multiplied by the variable so we divide both sides by 3.

[math]\begin{align} 3x &= 6 \\ \frac{3x}{3} &= \frac{6}{3} \\ 1*x &= 2 \qquad \text{we usually skip writing out this step} \\ x &= 2 \\ \end{align}[/math]

So, the first thing we do when solving these equations is to locate the integer on the same side of the equal sign as the variable, which is not multiplied by the variable. We combine that integer with its opposite in sign, which gives us 0. (Remember, that whatever operation you perform on one side of the equation, you must also perform to the other side.) The last thing we do is to divide both sides of the equation by the exact integer--including its sign-- that is multiplied by the variable.

Example 2:    [math]-2x -5 = 7[/math]

Step 1: There is a -5 with the -2x, so we combine it with a positive 5, and then also combine the 7 on the other side with a positive 5.

[math]\begin{align} -2x -5 &= 7 \\ -2x -5 +5 &= 7 + 5 \\ -2x &= 12 \\ \end{align}[/math]

Step 2: In order to solve for x, we want to get to 1*x which equals x. Therefore, we need to divide the -2 by -2 ( -2/-2) to get the 1. (Note: if we were to divide by 2, we would have -2/2 which is -1, and we would have -1*x. This would add another step in solving for x.)

[math]\begin{align} -2x &= 12 \\ \frac{-2x}{-2} &= \frac{12}{-2} \\ x &= -6 \\ \end{align}[/math]

Example 3:    [math]40 = -2 + 7x[/math]

Step 1: The variable is on the right side of the equation. There is a -2 added to the 7x, so we will add a positive 2 to both sides of the equation to cancel out that -2.

[math]\begin{align} 40 &= -2 + 7x \\ 40 + 2 &= -2 + 2 +7x \\ 42 &= 7x \\ \end{align}[/math]

Step 2: 7 is multiplied by the variable, so we divide both sides by 7.

[math]\begin{align} 42 &= 7x \\ \frac{42}{7} &= \frac{7x}{7} \\ 6 &= x \quad \text{or} \quad x = 6 \\ \end{align}[/math]

Example 4:    [math]-8y - 12 = 5[/math]

Step 1: There is a -12 with the -8y so we add 12 to both sides.

[math]\begin{align} -8y - 12 &= 5 \\ -8y - 12 + 12 &= 5 + 12 \\ -8y &= 17 \\ \end{align}[/math]

Step 2: -8 is multiplied by the variable, y, so we divide both sides of the equation by -8.

[math]\begin{align} -8y &= 17 \\ \frac{-8y}{-8} &= \frac{17}{-8} \\ y &= - \frac{17}{8} \qquad \text{a positive divided by a negative is a negative} \\ \end{align}[/math]