Solving Two-Step Linear Equations

While you can sometimes solve linear equations in a single step, it's more common to need two or more separate steps to solve for x. For example, the equation $3 x + 5 = 11$ calls for the value of x to be multiplied by 3 and then increased by 5. Solving this equation entails using inverse operations to undo each operation in reverse order, ultimately trying to isolate the x on one side of the equals sign.

Starting with $3 x + 5 = 11$ , the first step is subtracting 5 from each side:

$3 x = 6$

Next, we want to divide each side by 3 to get x by itself:

$x = 2$

We've solved the equation. As long as you remember to use the appropriate inverse operations (addition to undo subtraction, subtraction to undo addition, multiplication to undo division, and division to undo multiplication), and add/subtract before you multiply/divide, these problems can be pretty fun!

Why is this solving two-step “linear” equations?

Equations like the one above are considered "linear" because $x^{1}$ is the highest exponent present: there are no $x^{2}$ , $x^{3}$ , or any other powers. If you want to learn how to work with equations involving exponents, you'll want quadratic equations and polynomials.

Linear equations can also have more than one variable. For example, consider $y = 3 x + 2$ . You cannot solve an equation like this the way we did above because there are an infinite number of solutions based on different values for x and y. However, you can graph the equation as a line on the plane. Setting y to 0 and solving for x will reveal the coordinates of the y-intercept and help you graph it. Likewise, setting x to 0 and solving for y gives you the x-intercept. These are not the only solutions to the equation, but they will give you the two points you need to sketch the line.

Solving two-step linear equations practice questions

a. $5 x + 5 = 30$

To solve for x, first subtract 5 from both sides:

$5 x = 25$

Now, divide both sides by 5:

$x = 5$

b. $7 x - 1 = 27$

To solve for x, first add 1 to both sides:

$7 x = 28$

Now, divide both sides by 7:

$x = 4$

c. $\frac{x}{4} + 2 = 18$

To solve for x, first subtract 2 from both sides:

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

Now, multiply both sides by 4:

$x = 64$

d. $\frac{x}{3} - 7 = 8$

To solve for x, first add 7 to both sides:

$\frac{x}{3} = 15$

Now, multiply both sides by 3:

$x = 45$

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