Grade by Grade
C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Solving Equations with Integers and Decimals

C2.3 A1About 50 minutes
Learning goals
  • I can solve equations with multiple terms that involve integers and decimal numbers.
  • I can combine like terms with negative coefficients before solving.
  • I can verify a solution by substituting it into the original equation.
  • I can write and solve an equation for a real situation.

Words to know

equation
A statement that two expressions are equal, like 3x − 4 = −10.
solve
Find the value of the variable that makes the equation true.
solution
The value that makes an equation true. For 3x − 4 = −10, the solution is x = −2.
inverse operation
An operation that undoes another. Adding undoes subtracting; dividing undoes multiplying.
verify
Check a solution by substituting it into the original equation.

Remember from Grade 7

In Grade 7 you solved equations with multiple terms and decimals, like 2x + x + 4.5 = 13.5. You simplified each side, undid adding or subtracting, then undid multiplying or dividing, and verified.

In Grade 8 the numbers can be negative integers as well as decimals, and the solutions can be negative. The steps stay the same. You just need careful integer arithmetic.

Let's start on a frosty morning

At 7 a.m. it was −14°C. The temperature rose 2.5°C each hour. At what time was it −1.5°C?

Let h be the hours after 7 a.m. Then −14 + 2.5h = −1.5.

  • Add 14 to both sides: 2.5h = 12.5.
  • Divide both sides by 2.5: h = 5.

It was −1.5°C 5 hours after 7 a.m., at noon. Verify: −14 + 2.5(5) = −14 + 12.5 = −1.5 ✓.

−14 + 2.5h = −1.5: five jumps of 2.5, so h = 5−14−12−10−8−6−4−20+2.5+2.5+2.5+2.5+2.57 a.m.: −14°Cnoon: −1.5°C
From 7 a.m. to noon, the temperature makes 5 jumps of 2.5°C, from −14°C up to −1.5°C. That is why h = 5.
3x − 4 = −10=Add 4 to both sides. Now 3x = −6=Share into 3 equal groups: x = −2=
Solving 3x − 4 = −10 with tiles. Adding 4 positive units to both sides makes zero pairs on the left and leaves 3x = −6 (four of the negative units on the right also disappear in zero pairs). Sharing 6 negative units among 3 x-tiles gives x = −2.

Steps for solving

  1. Simplify each side: combine like terms (watch the signs!).
  2. Undo adding or subtracting: do the opposite to both sides.
  3. Undo multiplying or dividing: do the opposite to both sides.
  4. Verify in the original equation.

Keep the equation balanced: whatever you do to one side, do to the other.

Dividing by a negative

If you get −3n = 12, divide both sides by −3, not 3: n = 12 ÷ (−3) = −4. Different signs give a negative quotient.

Worked example

Example 1: A negative constant

Solve 3x − 4 = −10.

  1. Add 4 to both sides: 3x = −6.
  2. Divide both sides by 3: x = −2.
  3. Verify: 3(−2) − 4 = −6 − 4 = −10 ✓.

x = −2

Worked example

Example 2: Negative coefficients and like terms

Solve −5n + 2n + 7 = 19.

  1. Combine like terms: −5n + 2n = −3n, so −3n + 7 = 19.
  2. Subtract 7 from both sides: −3n = 12.
  3. Divide both sides by −3: n = −4.
  4. Verify in the original: −5(−4) + 2(−4) + 7 = 20 − 8 + 7 = 19 ✓.

n = −4

Worked example

Example 3: Decimals

Solve 1.2y − 3.5 = −0.5.

  1. Add 3.5 to both sides: 1.2y = 3.
  2. Divide both sides by 1.2: y = 3 ÷ 1.2 = 2.5. (Think: 12 × 2.5 = 30, so 1.2 × 2.5 = 3.)
  3. Verify: 1.2(2.5) − 3.5 = 3 − 3.5 = −0.5 ✓.

y = 2.5

Worked example

Example 4: A negative decimal coefficient

Solve −0.4k + 6 = 8.4.

  1. Subtract 6 from both sides: −0.4k = 2.4.
  2. Divide both sides by −0.4: k = 2.4 ÷ (−0.4) = −6.
  3. Verify: −0.4(−6) + 6 = 2.4 + 6 = 8.4 ✓.

k = −6

Worked example

Example 5: A bank balance

Mia's bank account is overdrawn: it shows −$18 (she owes the bank $18). She deposits $7.50 each week. After how many weeks will her balance be $27?

  1. Let w be the number of weeks: −18 + 7.5w = 27.
  2. Add 18 to both sides: 7.5w = 45.
  3. Divide both sides by 7.5: w = 6.
  4. Verify: −18 + 7.5(6) = −18 + 45 = 27 ✓.

After 6 weeks

Hands-on activity

Mystery number equations

You need: your paper algebra tiles from the last lesson (or paper and pencil), and a partner.

  1. Your partner secretly picks an integer from −10 to 10, say x = −3.
  2. They build an equation by choosing a few terms, for example 2x + x − 5, and working out its value: 2(−3) + (−3) − 5 = −14. They write 2x + x − 5 = −14 for you.
  3. You solve it and show each step. Then verify and compare with the secret number.
  4. Switch roles. Try at least one equation with a decimal, like 0.5x + 2 = 0.5.

What to look for: each step is written on its own line, both sides change the same way, and every solution is verified in the original equation. (0.5x + 2 = 0.5 gives 0.5x = −1.5, so x = −3.)

Just for fun

Dear x,

Please stop asking us to find you. You're not lost, you're just on the other side of the equal sign.

Sincerely, Grade 8

Your turn

1

Solve x + 9 = 4.

2

Solve 4x = −28.

3

Solve 2x − 7 = −15.

4

Solve −3m + 5 = 20.

5

Solve 6p − 2p − 3 = 9.

6

Solve 0.5t + 1.25 = −0.75.

7

Solve −2.5a − 4 = 6.

More practice: warm-up

1

Solve n − 6 = −10.

2

Solve −2x = 14.

3

Solve x ÷ 3 = −5.

4

Solve y + 2.5 = 1.

5

Solve 5k = −0.5.

More practice: on your own

1

Solve 7x + 3 − 2x = −22 and verify.

2

Solve −4w + 1.6 = −6.4.

3

Solve 0.3z − 1.2z = 2.7.

4

A diver starts at −2.5 m (below the surface) and goes down 1.5 m each minute. After how many minutes is she at −16 m?

5

A hockey team's goal difference (goals for minus goals against) is −7. The team then wins several games in a row, each by 2 goals, so the goal difference goes up by 2 each game. After how many of these games will it be +5?

Challenge

1

Solve 3(x − 2) + x = −18. (Hint: use the distributive property first.)

2

Write an equation with at least three terms whose solution is x = −2.5. Solve it to show it works.

3

The sum of three consecutive integers is −42. Find them.

Check your understanding

Quiz

1.

Solve 2x + 5 = −7.

2.

Solve −4n = 20.

3.

What is a good first step to solve 3x − 8 = 10?

4.

Solve 0.5x − 1.5 = −4.

5.

Solve 6y − 9y = 12.

6.

How do you verify that x = −2 solves 5x + 3 = −7?

For parents and teachers

Big idea: C2.3 asks Grade 8 students to solve equations that involve multiple terms, integers and decimal numbers in various contexts, and to verify their solutions.

Common difficulties:

  • Sign errors when combining like terms (−5n + 2n = −7n). Model it with tiles or a number line.
  • Dividing by the wrong sign (−3n = 12 → n = 4).
  • Changing only one side of the equation.
  • Skipping the check. Verifying catches most sign errors.

How to help: Play the mystery number game: you pick the secret number and build the equation, and your child solves it. Insist on one step per line and a check at the end. This lesson builds A1 skills: perseverance and self-monitoring.