Solving for a Variable:

Solving for a Variable Using Addition, Subtraction, Multiplication, and Division with Fractions

Sometimes solving for a variable requires more than one step. This lesson demonstrates how to solve for a variable when there is addition or subtraction as well as multiplication involving fractions. It is important to understand that the concepts don’t change whether there are whole numbers, decimals, or fractions in the equation. The principles of how to solve for a variable are still the same.

Video Source (06:53 mins) | Transcript

Equations can be much more complicated than simply having two operations. The following video will show you how to solve equations with many steps.

Video Source (12:45 mins) | Transcript

Remember, break these large problems down step by step by the Order of Operations (PEMDAS), then undo these steps by going backward and doing the inverse operation to find out what the variable is. It is important that we do each of these operations to both sides of the equation.

Helpful tip: Rewrite the problem after you do each step so you don’t lose track of what has already been done.

Additional Resources

Khan Academy: Example of Two-step Equations (03:03 mins, Transcript)
Khan Academy: Linear Equations 2 (06:06 mins, Transcript)
Khan Academy: Equations with Parentheses (06:03 mins, Transcript)

Practice Problems

Solve for the following variables:

$\frac{3}{2}{\text{x}} + \frac{1}{4} = \frac{13}{4}$

$\frac{1}{3} - \frac{2}{3}{\text{x}} = 3$

$3=-2{\text{D}}+\frac{2}{3}$

$-\frac{2}{3}=\frac{2}{3}{\text{U}}{-}\frac{1}{2}$

$8 = (5{\text{x}} {-} 2) - 5$

$2\left ( \frac{{\text{F}}+1}{-2}\right )-8=-4$

$1.3{\text{d}} + 5.2 = -2.6$

Solutions

$-4$ (Written Solution)
One thing you may find helpful is to first change any subtraction to the addition of a negative number. This can help avoid losing the negative.

To solve for the variable ${\text{x}}$ we use the order of operations backwards, but first, let’s find our steps going forward through the order of operations assuming we had a specific number for ${\text{x}}$.

If we were going forward, our steps would go as follows:
- Parentheses: None this time
- Exponents: None this time
- Multiplication & Division: between the $\left (\frac{-2}{3} \right )$ and ${\text{x}}$ so, first multiply the $\left (\frac{-2}{3} \right ){\text{x}}$
- Addition & subtraction: Add $\frac{1}{3}$ to our previous answer.
Going backwards to solve for a variable:

Since we are solving for a variable, we do these steps in reverse.

Step 1: Do the reverse of adding $\frac{1}{3}$ to each side of the equation. In other words, subtract $\frac{1}{3}$ from each side or add a negative $\frac{1}{3}$ to each side.

On the left-hand side:

The additive inverses $\frac{-1}{3}$ and $\frac{+1}{3}$ add together to $0$. This leaves just $\left (\frac{-2}{3} \right ){\text{x}}$ on the left side.

On the right-hand side:

We have $3+\left ( \frac{-1}{3} \right )$

Adding fractions requires common denominators. The number 3 is a whole number, so it has an invisible denominator of 1. We can rewrite it as $\frac{3}{1}$.

The greatest common denominator between 1 and 3 is 3. Multiply $\frac{3}{1}$ by $\frac{3}{3}$ to get the common denominator 3.

When adding or subtracting fractions with common denominators, we simply add or subtract the numbers in the numerator. The denominator stays the same because that is the part of the fraction that tells us what sizes the pieces are.

$\left ( \frac{9}{3} \right )+\left ( \frac{-1}{3} \right )=\frac{8}{3}$

We subtract $9-1$. The negative on the $\left ( \frac{1}{3} \right )$ is the same as adding a negative 1 or subtracting 1.

This leaves us with $\left ( \frac{-2}{3} \right ){\text{x}}$ on the left and $\frac{8}{3}$ on the right.

Step 2: Multiply both sides by the multiplicative inverse of $\frac{-2}{3}$.

This will leave just $1{\text{x}}$ on the left-hand side.

$-\frac{24}{6}=-4$

Our final solution: ${\text{x}} = -4$

$\frac{-7}{6}$

$\frac{-1}{4}$ (Solution Video | Transcript)

$-5$ (Solution Video | Transcript)

$-6$ (Written Solution)

We use the order of operations backwards to unravel this equation and solve for the variable d. This example uses decimal numbers. We treat them the same as integers or fractions.

Step 1: Do the opposite of any addition or subtraction (S&A in PEMDAS)

In this example, we need to add the additive inverse of positive 5.2 to both sides.

The additive inverse is $-5.2$, so we need to add $-5.2$ to both sides of the equation.

The left side of the equal sign:

$5.2 + -5.2 = 0$ leaving $1.3{\text{d}}$ on the left.

The right side of the equal sign:

$-2.6 + -5.2 = -7.8$

Remember the rules of adding a negative and a negative. They just become more negative.

Step 2: Do the opposite of any multiplication or division (D&M in PEMDAS)

In this example, we need to multiply both sides by the multiplicative inverse of $1.3$. The multiplicative inverse of a decimal is found the same way as the multiplicative inverse of an integer: $\frac{1}{\text{decimal}}$.

Multiply both sides by $\frac{1}{1.3}$
$This image shows two equations, one above the other. The top equation is one-point-three d equals (negative seven-point-eight). The negative is red to bring attention to it. The bottom equation is the same as the top equation except that both sides of the equal sign have the fraction (one-over-one-point-three) in blue multiplied what was originally there. In other words, (one-over-one-point-three) times one-point-three d equals (one-over-one-point-three) times (negative seven-point-eight).$
The left side of the equation:

$\left ( \frac{1}{1.3} \right )\left ( -1.3 \right ) = 1$ so we are left with $1{\text{d}}$.

The right side of the equation:

$\left ( \frac{1}{1.3} \right )\left ( -7.8 \right ) = \frac{\left ( -7.8 \right )}{\left ( 1.3 \right )}=-6$

We can solve this using longhand or a calculator.
$This image shows two equations, one over the other. The top equation is the same as the bottom equation of the previous image. It is (one-over-one-point-three) times one-point-three d equals (one-over-one-point-three) times (negative seven-point-eight). The fraction (one-over-one-point-three) on each side of the equal sign is blue. The bottom equation on the left-hand side shows a fraction whose numerator is one-point-three and denominator is a blue one-point-three to help show where the numbers went in the multiplication. This fraction is all multiplied to the variable d. The left-hand side is equal to the fraction negative seven-point-eight over one-point-three. Again the one-point-three is blue help identify where it came from.$
Our final solution: ${\text{d}} = -6$

Be sure not to lose the negative in the final solution.