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

Solving Equations with Adding and Subtracting

C2.2 A1About 40 minutes

Learning goals

I can think of an equation as a balance.
I can solve addition and subtraction equations with numbers up to 50.
I can use the opposite operation to solve an equation.
I can check (verify) my solution by substituting it into the equation.

Words to know

solve
Find the number that the variable stands for.

solution
The number that makes an equation true.

balance
When both sides of an equation have the same value.

opposite operation
The operation that undoes another one. Adding and subtracting undo each other.

verify
Check that a solution is correct by putting it back into the equation.

Remember

In the last lesson you wrote equations like c + 5 = 12. Now you will solve them: find the number the variable stands for. That number is called the solution.

An equation is like a balance

Think of a balance scale. When both sides weigh the same, the scale is level. An equation is the same: the equal sign means both sides have the same value.

The balance is level, so n + 12 has the same value as 30.

What number can go in n so that n + 12 balances 30? We need to find it, then check that the scale is still level.

Strategy 1: Think of a related fact

Ask: what number plus 12 makes 30? Count up from 12 to 30: 12 + 8 = 20, then 20 + 10 = 30. That is 8 + 10 = 18. So n = 18.

Strategy 2: Use the opposite operation

Adding 12 to n gave 30. To undo adding 12, subtract 12: 30 − 12 = 18. So n = 18.

Always verify

Put 18 back in place of n: 18 + 12 = 30. Both sides are 30. ✔ The scale balances, so n = 18 is the solution.

n + 12 = 30. Going forward, n adds 12 to make 30. To go back to n, undo it: 30 − 12 = 18.

If the equation is like ...	Undo it with ...	Example
n + 7 = 20	subtract 7	n = 20 − 7 = 13
n − 9 = 25	add 9	n = 25 + 9 = 34
15 + n = 41	subtract 15	n = 41 − 15 = 26
40 − n = 28	think: 40 take away what makes 28?	n = 40 − 28 = 12

Adding and subtracting undo each other.

Worked example

Example 1: An addition equation

Solve n + 17 = 45.

n plus 17 makes 45. Undo adding 17 by subtracting 17.
45 − 17: 45 − 10 = 35, then 35 − 7 = 28.
Verify: 28 + 17 = 45 ✔

n = 28

Worked example

Example 2: A subtraction equation

Solve m − 9 = 16. (This is Liam's equation from the last lesson.)

Liam had m dollars, spent 9 and had 16 left.
Undo subtracting 9 by adding 9: 16 + 9 = 25.
Verify: 25 − 9 = 16 ✔

m = 25. Liam had $25 at first.

Worked example

Example 3: The variable is being subtracted

Solve 50 − k = 32.

50 take away some number leaves 32. What did we take away?
Count up from 32 to 50: 32 + 8 = 40, then 40 + 10 = 50. That is 8 + 10 = 18.
Or think: the part taken away = 50 − 32 = 18.
Verify: 50 − 18 = 32 ✔

k = 18

Worked example

Example 4: The equal sign is on the other side

Solve 33 = p + 14.

The equal sign can be on either side. This says 33 is the same as p + 14.
Undo adding 14: 33 − 14 = 19.
Verify: 19 + 14 = 33 ✔

p = 19

Worked example

Example 5: Guess, check and revise

Solve 26 + g = 44 using guess and check.

Guess g = 20: 26 + 20 = 46. Too big by 2.
Revise: try g = 18: 26 + 18 = 44. ✔

g = 18. Guessing is fine, as long as you check and revise your guess.

A balance always stays level

If you do the same thing to both sides of a balance, it stays level. For n + 12 = 30, take 12 away from both sides: n = 30 − 12. That is why the opposite operation works!

Just for fun

Why is a balance scale such a fair referee?

It never takes sides. Well... it takes both sides, equally!

Hands-on activity

Kitchen balance equations

You need: a small cup or bag (the mystery cup), about 50 dried beans, pennies or paper clips, and paper. A real kitchen scale or a home-made coat-hanger balance is a bonus, but not needed.

Your grown-up secretly puts some beans in the cup. Then they add 9 more beans. They tell you: Now the cup has 25 beans.
Write the equation: b + 9 = 25.
Solve it on paper. Then empty the cup, count, and verify. Was b = 16?
Try a subtraction one: the grown-up puts some beans in, takes 7 out, and tells you 18 are left. Write b − 7 = 18 and solve.
Switch roles and make up equations for your grown-up.

What to look for: your child writes the equation first, uses the opposite operation, and checks by substituting.

Grown-up note: keep small items like these away from babies and toddlers.

Your turn

Solve n + 8 = 23.

Solve k − 12 = 20.

Solve 19 + a = 47.

Solve 36 − t = 21.

Solve 40 = w + 25.

Sara says the solution to y − 6 = 14 is 8. Verify. Is she right?

Write and solve: A number plus 27 is 50.

More practice: warm-up

Solve n + 5 = 12.

Solve m − 4 = 10.

Solve 10 + c = 30.

Is x = 6 the solution to x + 9 = 15?

What operation undoes subtracting?

More practice: on your own

Solve d + 26 = 49.

Solve h − 18 = 29.

Solve 45 − r = 17.

A rink had some skaters. 14 more came, and then there were 41. Write and solve an equation.

Owen had 50 stickers. He gave some away and has 36 left. Write and solve an equation.

Challenge

Solve n + 13 = 20 + 15.

Solve a + a = 38.

Mila solved k − 15 = 30 by doing 30 − 15 = 15. Explain her mistake and fix it.

Check your understanding

Quiz

What is the solution to n + 14 = 32?

What is the solution to p − 11 = 25?

How do you verify a solution?

What is the solution to 42 − k = 30?

What is the solution to 50 = 23 + m?

Which equation matches: Some apples were in a bowl. I ate 4. Now there are 9.

For parents and teachers

Big idea: The Ontario curriculum (C2.2) asks Grade 4 students to solve equations that involve whole numbers up to 50 in various contexts, and verify solutions. This lesson covers addition and subtraction equations; the next lesson covers multiplication and division.

Common difficulties:

Using the same operation instead of the opposite (k − 15 = 30 → 30 − 15). Ask: "What was done to k? What undoes it?"
Equations like 50 − k = 32, where the variable is subtracted. Here the answer is 50 − 32, not 50 + 32. Counting up from 32 to 50 helps.
Skipping the check. Make verifying a habit: substitute and see if both sides match.
Equal sign on the left (33 = p + 14). Remind your child the equal sign means "the same as", so it can be on either side.

How to help: Use real objects: hide some items in a cup, add more, say the total, and ask how many were hidden. Guessing and revising is a great strategy, not cheating (Strand A: problem solving and perseverance).