C. Algebra · Patterns, Expressions, Equations, Inequalities, Coding and Modelling

Solving Equations with Multiple Terms

C2.3 A1About 40 minutes

Learning goals

I can write an equation with more than one variable term to match a situation.
I can add like terms to make an equation simpler.
I can solve the equation by undoing operations, keeping it balanced.
I can verify my solution by substituting it back into the equation.

Words to know

equation
A statement that two expressions have the same value, like 3x + 2x = 45.

solve
Find the value of the variable that makes the equation true.

solution
The value that makes the equation true.

verify
Check a solution by substituting it into the original equation.

inverse (opposite) operation
An operation that undoes another: − undoes +, and ÷ undoes ×.

multiple terms
More than one term on a side of the equation, like 2n + n + 5.

Remember from Grade 5

In Grade 5 you solved equations with whole numbers up to 100, like 4 × n + 7 = 35. You thought of an equation as a balance, used opposite operations to undo steps, and verified your answer.

In Grade 6 your equations will have multiple terms, like 2n + n + 5 = 26. The new first step is to add the like terms (just like in the monomials lesson).

Let's start with a balance

On a balance scale, both sides have the same mass. An equation works the same way: both sides have the same value.

A balance scale. The left pan holds 2n + n + 5 and the right pan holds 26. The beam is level, because both sides have the same value.

Steps for solving

Simplify: add the like terms. 2n + n + 5 = 26 becomes 3n + 5 = 26.
Undo adding or subtracting: take 5 from both sides. 3n = 21.
Undo multiplying or dividing: divide both sides by 3. n = 7.
Verify in the original equation: 2 × 7 + 7 + 5 = 14 + 7 + 5 = 26 ✔.

Why does this work? If you take the same amount off both pans, the balance stays level.

Worked example

Example 1: Two like terms

Solve 3x + 2x = 45.

Simplify: 3x + 2x = 5x, so 5x = 45.
Undo × 5: 45 ÷ 5 = 9, so x = 9.
Verify: 3 × 9 + 2 × 9 = 27 + 18 = 45 ✔.

x = 9

Worked example

Example 2: Like terms and a number

Solve 4m + 6 + 2m = 60.

Simplify: 4m + 2m = 6m, so 6m + 6 = 60.
Undo + 6: 60 − 6 = 54, so 6m = 54.
Undo × 6: 54 ÷ 6 = 9, so m = 9.
Verify: 4 × 9 + 6 + 2 × 9 = 36 + 6 + 18 = 60 ✔.

m = 9

Worked example

Example 3: A shopping story

Liam buys 3 notebooks, then goes back for 2 more of the same notebooks. He also buys a $4 pen. He spends $19 in all. How much is one notebook?

Let n be the price of one notebook. Equation: 3n + 2n + 4 = 19.
Simplify: 5n + 4 = 19.
Undo + 4: 5n = 15. Undo × 5: n = 3.
Verify: 3 × 3 + 2 × 3 + 4 = 9 + 6 + 4 = 19 ✔.

One notebook costs $3.

A bar model of 3n + 2n + 4 = 19. Five equal n-bars plus the $4 pen make $19, so the five bars together are $15 and each is $3 (bars drawn to scale).

Worked example

Example 4: A perimeter equation

A triangle has two equal sides of length x and a third side of 10 cm. Its perimeter is 36 cm. How long are the equal sides?

Equation: x + x + 10 = 36.
Simplify: 2x + 10 = 36.
Undo + 10: 2x = 26. Undo × 2: x = 13.
Verify: 13 + 13 + 10 = 36 ✔.

Each equal side is 13 cm.

Worked example

Example 5: Guess, check and improve

Solve 4y + y + 8 = 48 with guess and check. Then solve it with the steps.

Guess y = 5: 20 + 5 + 8 = 33. Too low.
Guess y = 10: 40 + 10 + 8 = 58. Too high. So y is between 5 and 10.
Guess y = 8: 32 + 8 + 8 = 48 ✔.
With the steps: 5y + 8 = 48, so 5y = 40, so y = 8. Same answer!

y = 8. Guess and check works, but the steps are faster when the numbers get big.

Solve, then verify

Simplify first: add like terms on the side with the variable.
Undo in reverse: undo + or − first, then × or ÷.
Whatever you do to one side, do to the other side.
Verify in the original equation, not the simplified one. That way you catch slips in your simplifying, too.
If your check fails, don't erase everything. Look for the step where it went wrong.

Just for fun

Joke: Why is the equal sign so humble?

Because it knows it isn't less than or greater than anyone else.

Hands-on activity

Mystery envelopes

You need: 5 envelopes (or small cups turned upside down), about 40 paper clips (or dried beans), paper and a pencil.

Grown-up: secretly put the same number of paper clips in each envelope (for example, 5 in each). Seal or cover them.
Put 3 envelopes in one group and 2 envelopes in another group, then add some loose clips (for example, 4).
Tell your child the total number of clips (here: 3 × 5 + 2 × 5 + 4 = 29).
Child: write an equation, like 3e + 2e + 4 = 29. Simplify and solve it. Then open the envelopes to verify!
Swap roles: the child makes a mystery for the grown-up.

What to look for: 5e + 4 = 29, so 5e = 25, so e = 5. Your child should subtract the loose clips before dividing.

Grown-up note: keep paper clips and beans away from babies and toddlers.

Your turn

Solve 4x + 3x = 56.

Solve 2n + 3n + 10 = 45.

Solve y + y + y + 6 = 30.

Solve 6a + 2a − 4 = 60.

A rectangle has a length of 3w and a width of w. Its perimeter is 64 cm. Find w and the length.

Hana buys 2 bags of birdseed, then 1 more bag, all the same price. She also buys a $5 feeder. She pays $41. How much is one bag?

Is x = 6 a solution of 5x + 2x + 3 = 45?

More practice: warm-up

Solve x + 7 = 20.

Solve 4n = 36.

Simplify 2x + 5x.

Solve 3y = 27.

Is n = 5 a solution of 2n + 3 = 13?

More practice: on your own

Solve 5k + 4k + 12 = 84.

Solve 7m + 3 + m = 75.

A triangle has sides x, 2x and 2x. Its perimeter is 45 cm. Find each side.

A family rents toboggans for 4 children and 2 adults. Each toboggan costs the same. They also pay $6 for parking. The total is $48. What does one toboggan rental cost?

Jo solved 3x + 2x + 4 = 29 like this: 5x + 4 = 29, so 5x = 33, so x = 6.6. Find the mistake and fix it.

Challenge

Three whole numbers in a row (like 4, 5, 6) add up to 72. Use an equation to find them.

Sam has twice as many hockey cards as Ella. Kai has 5 more than Ella. Together they have 65 cards. How many does each person have?

Write your own equation with at least three terms whose solution is x = 6. Write a story to match.

Check your understanding

Quiz

Solve 4x + 2x = 42.

What is a good first step for 3n + 2n + 6 = 31?

Solve y + y + 8 = 30.

Which value is the solution of 5x + x = 54?

A triangle has two sides of length x and one side of 7 cm. The perimeter is 31 cm. What is x?

Why do we verify a solution?

For parents and teachers

Big idea: Grade 6 students solve equations that involve multiple terms and whole numbers in various contexts, and verify their solutions (C2.3). The new step is to add like terms first: 2n + n + 5 = 26 becomes 3n + 5 = 26, then 3n = 21, then n = 7.

Common difficulties:

Dividing before subtracting (26 ÷ 3 first). Remind your child to undo + and − before × and ÷.
Doing an operation to only one side. Use the balance picture: both pans must change the same way.
Verifying in the simplified equation instead of the original one.

How to help: Play "mystery envelopes" with paper clips or beans. Make up shopping stories with equal prices and a fixed extra (like a bag fee) and have your child write and solve the equation. Social-emotional learning (A1): guess and check is a perfectly good strategy to start with. Celebrate when your child notices the steps are faster.