6th Grade · 6.EE.B.7

One-step equations in 6th grade

One-step equations are equations that take a single move to solve, such as $x + 8 = 21$ or $6x = 54$ . In 6th grade, your child solves them with whole numbers, decimals and fractions. Your child also writes one-step equations from word problems.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

What it means, in plain words

An equation says that two amounts are equal. The letter, often $x$ , is a variable: it stands for the unknown number. Solving the equation means finding the number that makes both sides equal. A one-step equation needs only one move to get $x$ alone.

Solving one-step equations uses inverse operations, the operations that undo each other. Subtraction undoes addition, and division undoes multiplication. For $x + 8 = 21$ , subtract 8 from both sides to get $x = 13$ . For $6x = 54$ , divide both sides by 6 to get $x = 9$ . Whatever happens to one side happens to the other, so the two sides stay equal. One-step equations in 6th grade also use decimals and fractions, such as $x + 1.25 = 4$ or $\frac{1}{2}x = 7$ .

One-step equations word problems ask your child to write the equation first. The key question is what the story does to the unknown amount. If it adds a known amount, the equation looks like $x + 8 = 21$ . If it repeats the unknown amount several times, the equation looks like $6x = 54$ .

Your child should be able to:

Solve equations like $x + 8 = 21$ by subtracting the same number from both sides
Solve equations like $6x = 54$ by dividing both sides by the same number
Solve one-step equations with decimals and fractions
Write an equation for a word problem and check the answer in the story

Example

Apples and oranges together weigh 5 pounds. The apples weigh 3.4 pounds. How much do the oranges weigh?

Let $x$ be the weight of the oranges in pounds. The oranges plus the apples make 5 pounds, so $x + 3.4 = 5$ .
Subtract 3.4 from both sides: $x = 5 - 3.4 = 1.6$ .
Check: $1.6 + 3.4 = 5$ .
The oranges weigh 1.6 pounds.

Example

Six tickets to a school play cost 45 dollars in all. Each ticket costs the same. How much is one ticket?

Let $t$ be the cost of one ticket in dollars. Six tickets cost $6t$ , so $6t = 45$ .
Divide both sides by 6: $t = 45 \div 6 = 7.5$ .
Check: $6 \times 7.5 = 45$ .
One ticket costs 7 dollars and 50 cents.

Where children usually get stuck

For $x + 9 = 23$ , your child adds and writes $x = 32$ .
The plus sign looks like an instruction to add. The goal is the opposite: undo the adding to get $x$ alone.
What helps. Have your child put the answer back in: $32 + 9 = 41$ , not 23. Then ask what number plus 9 makes 23.
For $8x = 2$ , your child writes $x = 4$ .
Your child is used to dividing the bigger number by the smaller one, so $8 \div 2$ feels right. The equation asks for $2 \div 8$ .
What helps. Check by putting 4 back in: $8 \times 4 = 32$ , not 2. Then divide both sides by 8 to get $x = 0.25$ , and check again: $8 \times 0.25 = 2$ .
Your child solves $x + 8 = 21$ in the head and skips the written step, then gets stuck once decimals and fractions appear.
Small whole numbers can be solved by guessing, so writing the step feels pointless until the numbers get harder.
What helps. Ask for the written step even on easy problems: "subtract 8 from both sides". The same step solves $x + 1.25 = 4$ , where guessing is hard.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Each box holds 3 crayons. There are 12 crayons in all. Use x for the number of boxes.
Drag the number tiles into the blanks to complete both lines.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
x
=
x
=
36
9
3
15
12
4
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
Answer. 4
The answer is 4 because 12 divided by 3 is 4, so x = 4 and there are 4 boxes.
1. 13 crayons are in each box, so 3 is the coefficient. The total is 12.
2. 2x is the number of boxes, so divide both sides by 3.
3. 3After dividing, x = 4. So there are 4 boxes.
Common mistake. A common mistake is swapping 3 and 4 to make 4x = 12. It does not match the story because 3 is the number in each box and x is the number of boxes.
The idea underneath. The number next to x is the per-unit amount in the story (here, crayons per box), and solving px = q means dividing the whole amount by that coefficient.

2On level

Look at the student's solution to the equation below.

Equation	$\displaystyle x + 1.5$	$\displaystyle =$	$\displaystyle 7.3$
Student step 1	$\displaystyle x$	$\displaystyle =$	$\displaystyle 7.3 - 1.5$
Student step 2	$\displaystyle x$	$\displaystyle =$	$\displaystyle 5.2$

Select the statement that best describes the student's work.

The student used the wrong inverse operation.

There is no mistake; the solution is correct.

The student set up subtraction correctly but made an arithmetic error.

The student copied a number from the equation instead of solving.

Show the answer and how to solve it

The correct choice is that the student set up subtraction correctly but made an arithmetic error. The correct answer is x = 5.8, not x = 5.2.

1Read the student's work. The original equation is x + 1.5 = 7.3.
2Check the operation. To undo + 1.5, subtract 1.5 from both sides. Student step 1 shows x = 7.3 − 1.5, so the inverse operation is correct.
3Now compute the subtraction carefully.
4Student step 2 says x = 5.2, but the correct value is x = 5.8. The operation was set up correctly; the arithmetic contains an error.

Common mistake. A common mistake is checking only whether subtraction was chosen and not verifying the computation. Here, 7.3 − 1.5 = 5.8, not 5.2.

The idea underneath. When you analyze a solved equation, check two things in order: did the student choose the correct inverse operation, and did they compute that operation correctly?

3On level
Solve the equation.
$\displaystyle x + 2.47 = 3.82$
Enter only the value of x.
Loading...
Show the answer and how to solve it
Answer. 1.35
The answer is 1.35, because $1.35 + 2.47 = 3.82$ . Enter just 1.35.
1. 1Look at the equation. x is the missing number added to 2.47 to make 3.82.
2. 2Subtract 2.47 from both sides to undo the addition.
3. 3The left side simplifies to x.
4. 4Subtract the decimals: 3.82 − 2.47 = 1.35.
Common mistake. A common mistake is adding 2.47 and 3.82 instead of subtracting. Because 2.47 is added to x, you undo it by subtracting 2.47 from 3.82.
The idea underneath. Subtract the added number from both sides to leave x alone. Subtracting the same amount from equal values keeps them equal.
4Stretch
Look at the balance model below.
Drag the tiles to write a multiplication equation that matches the balance. Put the variable term on the left. Do not solve for the variable.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
÷
n
17.5
5
+
=
3.5
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
The answer is 5n = 17.5 because 5 equal n shapes balance 17.5.
1. 1The balance model shows 5 squares labeled n on the left and 17.5 on the right, so both sides have the same value.
2. 2Five copies of n can be written as 5n.
3. 3So the matching equation is 5n = 17.5.
Common mistake. Writing 5 + n = 17.5 adds 5 instead of showing 5 copies of n. Use 5n for those copies. The equation n = 3.5 gives the correct value of n, but this task asks for the multiplication equation shown by the balance.
The idea underneath. Repeated copies of the same variable are written with a coefficient, so 5 copies of n become 5n.