What is a two-step equation?
A two-step equation needs exactly two inverse operations to isolate the variable. It usually looks like ax + b = c, such as 3x + 5 = 20, where x has been multiplied and then had something added.
A one-step equation needs just one move. x − 4 = 10 only needs 4 added to both sides (x = 14), and 5x = 35 only needs a division by 5 (x = 7). The calculator handles both kinds.
How do you solve 3x + 5 = 20?
Subtract 5 from both sides, then divide both sides by 3. The answer is x = 5.
It helps to say each step out loud the first few times. "x was multiplied by 3, then 5 was added. So I take away the 5, then divide by 3." Once the order feels automatic, you can do most two-step equations in your head and use the calculator only to check.
Why do you undo addition or subtraction first?
Because solving runs the order of operations in reverse. To build 3x + 5 from x, you multiply by 3 first and add 5 last, so the 5 is the first thing to come off.
Think of getting dressed: socks go on before shoes, so shoes come off first. You can divide first if you like, but then you have to divide every term, giving x + 5/3 = 20/3. That works, it's just messier.
The same reverse-order idea carries into longer equations. Whatever was done to x last comes off first, which is why the solver always clears added or subtracted numbers before it divides.
| Building 3x + 5 | Solving 3x + 5 = 20 |
|---|---|
| 1. Multiply x by 3 | 1. Subtract 5 |
| 2. Add 5 | 2. Divide by 3 |
How do you solve a two-step equation with division?
Same order: undo the addition or subtraction, then multiply to undo the division. For x/4 − 2 = 6, add 2 to get x/4 = 8, then multiply both sides by 4 to get x = 32.
The solver writes x/4 as (1/4)x and divides by 1/4, which is the same as multiplying by 4. The steps don't change when x is on the right. In 7 = x/3 + 4, subtract 4 to get 3 = x/3, then multiply by 3 for x = 9.
What about negative coefficients?
Keep the negative sign attached to the number in front of x, and divide by that negative number in step 2. For −2x + 5 = 13, subtract 5 to get −2x = 8, then divide by −2 to get x = −4.
Watch equations where the constant comes first. In 5 − 2x = 11 the coefficient is −2, not 2. Subtract 5 to get −2x = 6, then divide by −2 for x = −3.
A bare −x means −1 times x. If you reach −x = 6, divide (or multiply) by −1 and you get x = −6.
A negative answer is perfectly normal, and so is a negative number on the right side. In −5x − 2 = 13, add 2 to get −5x = 15, then divide by −5 for x = −3. Check it: −5(−3) − 2 = 15 − 2 = 13.
What mistakes do students make with two-step equations?
Dividing before subtracting, and only dividing part of the equation, is the most common. From 3x + 5 = 20, writing x + 5 = 20/3 leaves the 5 undivided.
Dropping the negative is next. −2x = 8 gives x = −4, and writing x = 4 fails the check: −2(4) + 5 = −3, not 13.
With division equations, some students divide by 4 instead of multiplying. x/4 = 8 means x is 4 times bigger than 8, so x = 32, not 2.
How do you solve a one-step equation?
Use the one inverse operation that undoes what was done to x. Each of the four basic types has its own move.
| Equation | Move | Answer |
|---|---|---|
| x + 7 = 3 | Subtract 7 from both sides | x = −4 |
| x − 9 = −2 | Add 9 to both sides | x = 7 |
| 4x = −20 | Divide both sides by 4 | x = −5 |
| x/5 = 3 | Multiply both sides by 5 | x = 15 |
A two-step equation is two of these stacked together. If one-step equations still feel shaky, practice those first, because every mistake there carries into the longer problems. Pay extra attention to rows like x + 7 = 3, where the answer comes out negative even though every number in the problem is positive.
What does a two-step equation look like in a word problem?
Most of them have a starting amount plus a rate. Suppose you have $5 and save $3 each week. To find how many weeks until you have $20, write 3x + 5 = 20.
That's the same equation as the worked example, so x = 5 weeks. The 5 is the starting amount and the 3 is the rate. Spotting which number is which is most of the work, and once you have the equation the two steps take care of the rest.