Write, Solve and Justify One-Step Equations | Grade 6 Math | LevelKite

A story in words and a line of symbols can say exactly the same thing. A variable is a letter, such as $n$ , that stands for a number you do not know yet. An expression is a number, a variable, or several of them joined by operations, such as $n + 9$ . An equation is a statement that two expressions are equal, such as $n + 9 = 21$ .

You can already solve one-step addition and subtraction equations. They need just one operation to solve, and the numbers can be whole numbers, fractions or decimals. In this lesson you do four connected things. You write an equation from words. You use an equation to solve a word problem. You explain why each solving step is allowed. And you find the mistake in a solution that went wrong.

To write an equation from words, look for two amounts that the words say are equal. Then build the equation in four small moves:

Name the unknown. Choose a variable and say exactly what it counts, with its unit. Write “let $w$ be the number of liters of water,” not just “ $w$ is water.”
Say the relationship in words. Make a short sentence with “equals” or “is” in the middle.
Swap in the numbers and the variable. The part of the sentence that says the two amounts match becomes the equals sign.
Read the equation back. It should tell the same story as the words.

The table shows four sentences turned into equations.

From words to a one-step equation

Words	What the variable stands for	Equation
A number increased by 9 is 21.	$n$ is the unknown number.	$n + 9 = 21$
7 less than a number is 15.	$n$ is the unknown number.	$n - 7 = 15$
After Ana walks 2.4 kilometers of a trail, 1.1 kilometers are left.	$t$ is the length of the trail in kilometers.	$t - 2.4 = 1.1$
Pouring in $\frac{3}{4}$ cup of milk brings the jug to 2 cups.	$c$ is the number of cups of milk in the jug before pouring.	$c + \frac{3}{4} = 2$

Worked example

Write an equation from a comparison

Mia has 12 more stickers than Leo. Mia has 30 stickers. Write an equation you could solve to find how many stickers Leo has.

Name the unknown. The question asks about Leo, so let s be the number of stickers Leo has. Saying what s counts, and in what unit, keeps the equation tied to the story.
$s = \text{number of stickers Leo has}$
Say the relationship in words. Mia has 12 more than Leo, so Leo's count with 12 added on equals Mia's count.
$\text{Leo's stickers} + 12 = \text{Mia's stickers}$
Swap in what you know. Leo's count is s, and Mia's count is 30. Both sides count stickers, so the units match.
$s + 12 = 30$
Read the equation back against the story: Leo's stickers plus 12 make Mia's 30 stickers. The 12 is added to Leo's count because Leo is the one with fewer stickers.

Answer

The equation is $s + 12 = 30$ , where $s$ is the number of stickers Leo has.

Writing the equation is half the job. Next you solve it, which means finding the value of the variable that makes the equation true. That value is the solution. You have done this before. Now the goal is to see why the method works.

Picture the equals sign as the center of a balance. The two sides hold the same amount. If you add 5 to one side only, the balance tips. If you add 5 to both sides, it stays level. Solving uses this idea: change both sides in the same way, so that the variable ends up alone on one side.

Key concept

Do the same valid operation to both sides

An equation says that its two sides have the same value. If you add the same number to both sides, or subtract the same number from both sides, the sides still match. The new equation is true for the same value of the variable.

To get the variable alone, choose the inverse operation, the one that undoes what is done to the variable. Addition and subtraction of the same number undo each other: do one, then the other, and you are back at the starting number.

The operation must be valid. Adding or subtracting the same number on both sides always is. Dividing both sides is valid only when you divide by a number that is not zero. This lesson uses only addition and subtraction.

Below, $x$ is the variable, and $a$ and $b$ are known numbers.

\text{If } x + a = b \text{, then } x + a - a = b - a. \quad \text{If } x - a = b \text{, then } x - a + a = b + a.

Worked example

Solve a word problem with an equation

A baker opens a new bag of flour. She uses 3.75 kilograms and has 8.5 kilograms left. How many kilograms of flour were in the bag at the start?

Name the unknown with its unit. Let f be the number of kilograms of flour in the bag at the start.
$f = \text{kilograms of flour at the start}$
Write the relationship as an equation. The flour at the start, take away the flour used, equals the flour left. Every amount is in kilograms.
$f - 3.75 = 8.5$
Add 3.75 to both sides. Addition is the inverse of subtracting 3.75, and adding the same number to both sides keeps the sides equal.
$f - 3.75 + 3.75 = 8.5 + 3.75$
Simplify each side. On the left, subtracting 3.75 and then adding 3.75 brings you back to f. On the right, think of 8.5 as 8.50. Then 8 + 3 = 11 and 0.50 + 0.75 = 1.25, so the sum is 12.25.
$f = 8.5 + 3.75 = 12.25$
Check in the equation by replacing f with 12.25. The statement is true, so 12.25 is a solution.
$12.25 - 3.75 = 8.5$
Check against the story. A bag that starts with 12.25 kilograms and loses 3.75 kilograms has 8.5 kilograms left, just as the problem says.

Answer

The bag held $12.25$ kilograms of flour at the start.

Now look closely at the why. When you explain a solution, every line needs a reason. Solving a one-step addition or subtraction equation uses only two kinds of moves, so you need only two kinds of reasons:

You add the same number to both sides, or subtract the same number from both sides. Reason: when two equal amounts both gain, or both lose, the same number, they are still equal.
You rewrite one side. Reason: the new expression is equivalent to the old one. It has the same value for every value of the variable, so the equation still says the same thing.

A strong explanation also says why that operation was chosen. It is the inverse of the operation in the equation, so it leaves the variable alone. Back up each rewrite with a reason that works for every value. Trying one value is not enough to show that two expressions are equivalent.

Worked example

Explain each line of a solution

This solution is correct. The equation is $y + \frac{1}{4} = \frac{7}{8}$ , where $y$ is an unknown number. Line 1: $y + \frac{1}{4} - \frac{1}{4} = \frac{7}{8} - \frac{1}{4}$ . Line 2: $y = \frac{7}{8} - \frac{2}{8}$ . Line 3: $y = \frac{5}{8}$ . Explain why each line is allowed.

Line 1 subtracts one fourth from both sides. This is allowed because the two sides have the same value, and taking the same amount from each leaves them equal. Subtraction is the right choice because it is the inverse of adding one fourth.
$y + \frac{1}{4} - \frac{1}{4} = \frac{7}{8} - \frac{1}{4}$
Line 2 rewrites the left side as y alone. Adding one fourth and then subtracting one fourth is the same as adding 0, and adding 0 changes no number. So the old left side is equivalent to y: the two have the same value for every value of y.
$y + \frac{1}{4} - \frac{1}{4} = y + 0 = y$
Line 2 also renames one fourth on the right side as two eighths. Multiplying the numerator and the denominator by 2 gives an equal fraction, so the value of the right side does not change. You need eighths to do the subtraction.
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
Line 3 finishes the subtraction. The denominators match, so subtract the numerators: 7 eighths take away 2 eighths leaves 5 eighths. The right side is replaced by an equal number, so the equation still says the same thing.
$\frac{7}{8} - \frac{2}{8} = \frac{5}{8}$
Check by replacing y with five eighths in the original equation. The statement is true, so five eighths is a solution.
$\frac{5}{8} + \frac{1}{4} = \frac{5}{8} + \frac{2}{8} = \frac{7}{8}$

Answer

Line 1 subtracts the same number from both sides, and lines 2 and 3 only replace expressions with equal ones. So every line is true for the same value of $y$ , and that value is $\frac{5}{8}$ .

The same two reasons help you find a mistake in a solution. Work in this order:

Test the answer. Substitute it into the original equation. If the statement is false, the answer is wrong for certain.
Find the line where equality broke. Compare each line with the line before it. Ask what happened to the left side and what happened to the right side. A line is a mistake when it changes one side by a different amount than the other, or when it rewrites one side as something with a different value.
Fix that line and solve again. Then check the new answer in the original equation.

Worked example

Find and fix a mistake

Rico solved $p - 2.8 = 6.5$ , where $p$ is an unknown number. First he wrote $p = 6.5 - 2.8$ . Then he wrote $p = 3.7$ . Find Rico's mistake, explain why it is a mistake, and give the correct solution.

Test Rico's answer in the original equation by replacing p with 3.7. The left side comes to 0.9, not 6.5. One false statement is enough to prove that 3.7 is not a solution. It is also too small to make sense: taking 2.8 away from p leaves 6.5, so p must be greater than 6.5.
$3.7 - 2.8 = 0.9 \neq 6.5$
Find the line where equality broke. Compare the equation with Rico's first line. The left side goes from p minus 2.8 to just p. That change adds 2.8. The right side goes from 6.5 to 6.5 minus 2.8. That change subtracts 2.8.
$\text{left side: add } 2.8 \qquad \text{right side: subtract } 2.8$
Explain why this fails. The two sides started equal. Then one side went up by 2.8 and the other went down by 2.8, so they are no longer equal. Because the left side gained 2.8, the right side needs to gain 2.8 too.
Correct the step. Add 2.8 to both sides. Addition is the inverse of subtracting 2.8, and adding the same number to both sides keeps the sides equal.
$p - 2.8 + 2.8 = 6.5 + 2.8$
Simplify each side. On the left, subtracting 2.8 and then adding 2.8 brings you back to p. On the right, 6 + 2 = 8 and 0.5 + 0.8 = 1.3, so the sum is 9.3.
$p = 6.5 + 2.8 = 9.3$
Check the new answer in the original equation. The statement is true, so 9.3 is the solution.
$9.3 - 2.8 = 6.5$

Answer

Rico subtracted $2.8$ on the right side while the left side had $2.8$ added, so the two sides stopped being equal. Adding $2.8$ to both sides gives the correct solution, $p = 9.3$ .

Here is the idea to keep. An equation says that two amounts are equal. To write one from words, name the unknown with its unit, say the relationship in words, and swap in the numbers and the variable. To solve it, add or subtract the same number on both sides, choosing the inverse operation so the variable ends up alone. Every line of a solution has a reason: either you added or subtracted the same number on both sides, or you rewrote one side with an equal value. When a solution goes wrong, a false check tells you so, and comparing the two sides line by line shows you where.