Grade by Grade
C. Algebra · Patterns, Integers, Equations, Inequalities, Coding and Modelling

Solving Equations with Multiple Terms and Decimals

C2.3 A1About 50 minutes
Learning goals
  • I can simplify an equation by combining like terms.
  • I can solve equations with multiple terms, whole numbers and decimals by doing the same thing to both sides.
  • I can verify a solution by substituting it into the original equation.
  • I can write and solve equations for real-life problems.

Words to know

equation
A statement that two expressions are equal, like 3x + 4.5 = 13.5.
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 operations
Operations that undo each other: + and −, × and ÷.
balance
A model of an equation. Whatever you do to one side, you must do to the other to keep it level.

Remember from Grade 6

In Grade 6 you solved equations with multiple terms and whole numbers, like 2n + n + 5 = 26. First you added like terms (3n + 5 = 26), then you undid the steps: subtract 5, then divide by 3. You thought of an equation as a balance and verified your answer.

In Grade 7, the equations can also have decimal numbers, and the like terms can be subtracted as well as added.

Let's start with a puzzle

In the balance below, every x is the same unknown mass. The left side has 2x, another x and 4.5 kg. The right side has 13.5 kg. How much is x?

2x + x + 4.513.52x + x + 4.5 = 13.5Both sides are equal, so the balance is level.
A level balance. The left pan holds 2x + x + 4.5. The right pan holds 13.5.

Four steps to solve

  1. Simplify each side by combining like terms. 2x + x + 4.5 becomes 3x + 4.5.
  2. Undo adding or subtracting. Do the same thing to both sides. Subtract 4.5 from both sides: 3x = 9.
  3. Undo multiplying or dividing. Divide both sides by 3: x = 3.
  4. Verify in the original equation: 2(3) + 3 + 4.5 = 6 + 3 + 4.5 = 13.5 ✓.

Why that order? To build 3x + 4.5, you multiply x by 3 and then add 4.5. To undo it, go backwards: subtract first, then divide. It's like taking off your boots before your socks: the last thing on is the first thing off.

xxx4.513.53x + 4.5 = 13.5startxxx93x = 9take 4.5 off both sidesx3x = 3share each side by 3Whatever you do to one side, do to the other side too.(2x + x is already combined into 3x here.)
The balance stays level at every step because the same thing happens to both sides. Take away 4.5 kg from each pan, then split each side into 3 equal shares: x = 3.

Keep it balanced

Write each step on a new line, directly under the one before, with the equal signs lined up. Say what you're doing to both sides: "Subtract 4.5 from both sides."

Worked example

Example 1: Decimal coefficient

Solve 0.75n + 4.25 = 10.25.

  1. Subtract 4.25 from both sides: 0.75n = 6.
  2. Divide both sides by 0.75: n = 6 ÷ 0.75 = 8. (Think: how many 0.75s make 6? Four 0.75s make 3, so eight make 6.)
  3. Verify: 0.75 × 8 + 4.25 = 6 + 4.25 = 10.25 ✓.

n = 8

Worked example

Example 2: Subtracting like terms

Solve 5y − 2y + 1.5 = 10.5.

  1. Simplify: 5y − 2y = 3y, so 3y + 1.5 = 10.5.
  2. Subtract 1.5 from both sides: 3y = 9.
  3. Divide both sides by 3: y = 3.
  4. Verify in the original: 5(3) − 2(3) + 1.5 = 15 − 6 + 1.5 = 10.5 ✓.

y = 3

Worked example

Example 3: A class trip

A bus for a class trip to a sugar bush costs $180, plus $12.50 for each student's admission. The total is $555. How many students are going? (All prices in this lesson are made up for practice.)

  1. Let s be the number of students. Equation: 12.5s + 180 = 555.
  2. Subtract 180 from both sides: 12.5s = 375.
  3. Divide both sides by 12.5: s = 30.
  4. Verify: 12.5 × 30 + 180 = 375 + 180 = 555 ✓.

30 students are going.

Worked example

Example 4: Multiple terms in a story

Leo buys 3 pizzas on Friday and 2 more on Saturday, all the same price. He also buys a $4.75 drink. He spends $59.75 in all. What does one pizza cost?

  1. Let p be the price of one pizza. Equation: 3p + 2p + 4.75 = 59.75.
  2. Simplify: 5p + 4.75 = 59.75.
  3. Subtract 4.75 from both sides: 5p = 55. Divide by 5: p = 11.
  4. Verify: 33 + 22 + 4.75 = 59.75 ✓.

One pizza costs $11.

Worked example

Example 5: Division in an equation

Solve n ÷ 4 + 2.5 = 6.

  1. Subtract 2.5 from both sides: n ÷ 4 = 3.5.
  2. Undo dividing by 4: multiply both sides by 4. n = 14.
  3. Verify: 14 ÷ 4 + 2.5 = 3.5 + 2.5 = 6 ✓.

n = 14

Just for fun

How does x always keep its balance?

Whatever happens to one side happens to the other side too!

Hands-on activity

Mystery bags on a balance

You need: a few identical small paper bags or envelopes, some coins, paper and a pencil. (A homemade balance from a hanger, string and two cups works too, but drawings are fine.) Keep coins away from young children.

  1. A grown-up secretly puts the same number of coins in each of 3 envelopes, then sets 2 loose coins beside them. They count the total and tell you, for example, "3 envelopes and 2 coins make 20 coins."
  2. Write the equation: 3e + 2 = 20.
  3. Solve it on paper, saying "I do the same to both sides" at each step. (e = 6)
  4. Open one envelope to verify.
  5. Make it harder: make two groups of envelopes, like 2 envelopes and 1 more envelope, plus some loose coins, and write equations with multiple terms, like 2e + e + 5 = 23. (e = 6)
  6. Switch roles and make a puzzle for your grown-up.

Your turn

1

Solve 4x + 2x = 42.

2

Solve 3n + 5 = 26.

3

Solve 2.5m + 1.5 = 11.5.

4

Solve 6y − y + 3 = 28.

5

Solve 0.4k + 2.6 = 5.

6

Solve x ÷ 3 + 1.5 = 4.5.

7

Three movie tickets at the same price and a $6.75 popcorn cost $41.25. Write and solve an equation to find the price of a ticket.

More practice: warm-up

1

Solve x + 2.5 = 7.

2

Solve 3y = 1.8.

3

Solve n − 1.2 = 4.

4

Solve 2a + 3a = 20.

5

Solve 10w = 35.

More practice: on your own

1

Solve 7.5x + 2.5x − 4 = 26.

2

Solve 1.25b + 3.75 = 10.

3

Renting the arena costs $95 plus $3.50 per skater. The total bill is $172. How many skaters came?

4

A rectangle is 3x long and x wide. Its perimeter is 36.8 cm. Find x and the side lengths.

5

Is x = 2.5 a solution of 4x + 2x − 3 = 12? Show how you know.

Challenge

1

Solve 4x + 3 = x + 15. (Hint: on a balance, you can take one x off each side.)

2

Three numbers are n, n + 1.5 and n + 3. Their sum is 22.5. Find the numbers.

3

Write a real-life story for 2.5h + 10 = 25, then solve it.

Check your understanding

Quiz

1.

Solve 2x + 3x = 35.

2.

What is the first step to solve 4n + 2.4 = 14.4?

3.

Solve 0.5m + 1 = 4.

4.

Solve 6y − 2y + 5 = 25.

5.

Which value is a solution of 3x + 1.5 = 9?

6.

A rink charges $6 to enter plus $2.75 per skate rental hour. Ella pays $14.25. Which equation fits?

For parents and teachers

Big idea: Grade 7 students solve equations that involve multiple terms, whole numbers and decimal numbers in various contexts, and verify their solutions (C2.3). The steps are: combine like terms, undo addition or subtraction, undo multiplication or division, verify in the original equation.

Common difficulties:

  • Dividing before subtracting (14.4 ÷ 4 first). Use the boots-and-socks idea: undo the last step first.
  • Doing something to only one side. Draw the balance.
  • Dividing by a decimal. For 6 ÷ 0.75, ask "How many 0.75s fit into 6?" or use money (how many 75¢ in $6?).
  • Verifying in the simplified equation instead of the original.

How to help: Play the mystery bags game. Ask your child to say each step out loud and to always verify. Praise persistence when an equation takes several steps. This lesson builds A1 skills: perseverance, organizing work clearly and checking answers.