Two-step equations

To solve 3x + 5 = 20, undo the + 5 first and the × 3 second: 3x = 15, then x = 5.

Two operations, undone in reverse order

Most equations need more than one move. In 3x + 5 = 20, two things have happened to x: it was multiplied by 3, and then 5 was added. To get x back you undo them in reverse order, the way you take off shoes before socks. First subtract 5 from both sides, which gives 3x = 15. Then divide both sides by 3, which gives x = 5. Check in the original equation: 3 × 5 + 5 = 20, and the right side is 20, so the answer fits.

Why undo the addition first? You could divide everything by 3 straight away, but then you would get x + 5/3 = 20/3, which is correct and messier. Removing the constant first leaves a single term with the variable, and one division finishes the job. The same order works for any equation of the form ax + b = c: subtract b, then divide by a. In symbols the answer is x = (c − b) ÷ a, though in practice it is easier to do the two steps than to remember the formula.

Signs and word problems

Negative numbers do not change the order, only the arithmetic. In −4y − 7 = 9, add 7 to both sides to get −4y = 16, then divide by −4 to get y = −4. Check: −4 × (−4) − 7 = 16 − 7 = 9. When the constant comes first, as in 12 − 2x = 4, it helps to read the equation as 12 + (−2)x = 4. Subtract 12 to get −2x = −8, then divide by −2 to get x = 4. Writing the sign together with its number keeps the steps honest.

Two-step equations come up whenever there is a fixed amount plus a rate. Here is a made-up example. A bike rental costs 5 dollars to start plus 3 dollars per hour, and you paid 20 dollars. If h is the number of hours, the cost is 3h + 5, so 3h + 5 = 20 and h = 5 hours. The fixed amount is the constant you remove first, and the rate is the coefficient you divide by last. Write the equation from the words before you calculate, and say what the variable means in your own words.

In the graph lab the blue line is y = 3x + 5 and the orange line is y = 20. Move b and the blue line slides up or down without changing its tilt; move a and it tilts around the point (0, b). Either way the crossing point is the solution of the two-step equation. Below the lab, the steps are written out so you can practise saying them in order.