Solving One-Step Linear Equations: Division

When you're evaluating a linear algebraic expression, you can sometimes get the answer by using a single operation (addition, subtraction, multiplication, or division). For these equations, you use the inverse operation, or the opposite of the operation in the equation. The inverse of addition is subtraction, and the inverse of subtraction is addition. Likewise, the inverse of multiplication is division, and the inverse of division is multiplication.

This can seem very counterintuitive in practice since you want to add as soon as you see a + sign, not subtract. That said, you'll get the hang of it with a little practice. Many students find working with division to be the most challenging, and the rest of this article will focus on it.

Exploring how to solve one-step linear equations with division

Since you use the inverse operation to solve one-step linear equations, division problems are approached through multiplication. Consider the following example:

Solve for x:

$\frac{x}{3}=5$

Remember that we can do whatever we like to one side of the equation, aside from dividing by zero, as long as we do the same thing to the other side. Since we want to get x alone, we multiply both sides by 3 to create an equivalent expression. That gives us:

$x=5\left(3\right)=15$

15 is our final answer! You might see several different variables in one-step linear equations, but the letter chosen doesn't actually affect how you solve a problem. For example, here's an example with an n:

Solve for n:

$\frac{n}{5}=25$

Once again, we want the n alone, so we multiply each side by 5. That means $n=5\left(25\right)=125$ , meaning we've solved for n the same way we did for x.

Similarly, there's more than one way to present a division problem. Consider the following example:

Solve for x:

$\frac{x}{3}=3$

It might look different, but we approach it in the exact same way. Multiply each side by 3 to get x on its own. That gives us $n=3\left(3\right)=9$ , meaning that we've once again solved the equation with a single step.

The most important thing is to not let anything throw you off. Whether division is expressed through a fraction or a division symbol, your first instinct should be to isolate the variable (whatever it happens to be) by multiplying both sides by a constant value.

Practice Questions

a. Solve for y: $\frac{y}{2}=8$

To solve for y, multiply both sides of the equation by 2:

$y=2\times 8$

$y=16$

b. Solve for x: $\frac{x}{3}=27$

To solve for x, multiply both sides of the equation by 3:

$x=3\times 27$

$x=81$

c. Solve for n: $\frac{n}{5}=10$

To solve for n, multiply both sides of the equation by 5:

$n=5\times 10$

$n=50$

d. Solve for x: $x\times 3=6$

To solve for x, divide both sides of the equation by 3:

$x=\frac{6}{3}$

$x=2$

e. Solve for n: $\frac{n}{3}=5$

To solve for n, multiply both sides of the equation by 3:

$n=3\times 5$

$n=15$

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Varsity Tutors can help with solving one-step linear equations with division

Solving one-step linear equations through division is the gateway to more advanced concepts in algebra such as solving equations with more than one step, working with polynomials, and even working with multiple variables at once. If your student is struggling now, the problem could snowball with time. Luckily, intervention in the form of an experienced 1-on-1 math tutor could equip your student with the foundational skills and self-confidence they need to approach all kinds of math problems successfully. Contact the Educational Directors at Varsity Tutors today for further information on private tutoring and to sign up.