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# Solving One-Step Linear Equations

It's possible to solve some linear equations in a single step using a single operation (addition, subtraction, multiplication, or division). In these scenarios, you use the inverse operation to solve it. The inverse of addition is subtraction, and the inverse of subtraction is addition. Similarly, multiplication is the inverse of division, and division is the inverse of multiplication.

The hardest part of solving these equations is often performing a different operation than the sign in the equation would suggest. For example, $n+8=10$ seems like an addition problem due to the plus sign but is actually solved by subtracting 8 from both sides. Practice is the best way to feel more comfortable with these problems, so let's take a look at examples for all four operations.

## Solving one-step linear equations (addition)

$x+7=13$

Just like the example above, we need to get x by itself to determine its value. Remember that you can do anything you want to an algebraic expression as long as you do it to both sides (aside from dividing by zero), so we subtract 7 from the left-hand side to isolate the x and then subtract 7 from the right side to maintain a mathematically-equivalent expression. That leaves us with $x=13-7=6$ , meaning that we have successfully solved for x!

## Solving one-step linear equations (subtraction)

$n-8\ge 14$

We're looking at a minus sign this time, but the procedure is still the same. We need to isolate the variable (in this case n), so we add 8 to both sides of the equation. That gives us a solution of n 22. Note that this is an inequality rather than a straight equation with an = sign, but that doesn't change how you should approach it.

## Solving one-step linear equations (multiplication)

$\frac{3}{4}y=15$

The inverse of multiplication is division, which means we want to divide to get y by itself. As you may recall from your knowledge of fractions, multiplying by $\frac{4}{3}$ is the same as dividing by $\frac{3}{4}$ . Therefore, we can multiply each side by $\frac{3}{4}$ to cancel out the fractions on the left side while changing the right side to $\frac{4}{3}\left(15\right)$ . From there, you can quickly see that $y=\frac{60}{3}=20$ .

That might've felt more like a fractions problem than a one-step linear equation, so here's another example:

$8x=24$

The question is asking "What number multiplied by 8 equals 24?" So we start by dividing each side by 8 which isolates x on the left side with $x=\frac{24}{8}=3$ .

## Solving one-step linear equations (division)

$\frac{x}{4}=16$

We need to multiply each side by 4 since multiplication is the inverse of division, which tells us that $x=64$ . Division equations could also look like this:

$\frac{n}{10}=10$

We need to multiply each side by 10 here, leaving us with $n=100$ . If you want to check your work, you can plug your answer into the original equation to verify you get a true statement. 100 divided by 10 equals 10, so that works out. If you got an answer of 88, 88 divided by 10 does not equal 10, so you would know that you made a mistake.

## Solving one-step linear equations practice questions

a. $17+x\le 23$

$x\le 6$

b. $37-x=21$

$x=16$

c. $9x=99$

$x=11$

d. $\frac{y}{5}=125$

$x=625$

e. $\frac{n}{3}=12$

$n=36$

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