Solve and verify multi-term whole-number equations

Ontario Grade 6 Mathematics

Algebra

Use equal amounts on both sides to find an unknown and check your answer.

Imagine two bags and some loose counters. The bags hold the same unknown number of counters. If both sides of a balance show the same total, they are equal. An equation is a number sentence that says two amounts are equal. A letter can stand for an unknown number. In this lesson, you will use whole numbers, which are numbers such as

0

,

1

, and

2

, to find the unknown and check that your answer works.

What you will learn

Explain what an equation and an unknown represent.
Solve multi-term whole-number equations by keeping both sides equal.
Verify a solution by putting it back into the original equation.

Start with equal amounts

An equation has an equals sign. The equals sign means that the amount on its left is the same as the amount on its right. It does not mean “the answer comes next.”

A term is one part of an expression, separated from another part by a plus or minus sign. For example,

4n+18

has two terms:

4n

and

18

. The letter

n

is an unknown. It stands for a number we are trying to find.

The number beside a letter tells how many equal groups of that number there are. So

4n

means four groups of

n

. If

n

is

3

, then

4n

is

4

groups of

3

, or

12

.

Use a balance idea to think about an equation. If you take away the same amount from both sides, or split both sides into the same number of equal groups, the two sides stay equal. These actions help undo an operation and reveal the unknown.

\text{left side}=\text{right side}

An equation says that two amounts are equal.
A letter can stand for an unknown whole number.
Keep both sides equal as you solve.

Undo operations while keeping equality

To solve an equation means to find the number that makes it true. Start by noticing what is being done to the unknown. Then undo those operations in a sensible order.

For an equation such as

4n+18=70

, the unknown is multiplied by

4

, and then

18

is added. First remove the added

18

from both sides. Then split the remaining amount into

4

equal groups. Each step keeps the two sides equal.

When an equation has unknown terms on both sides, collect the unknown terms on one side. For example, in

5a+12=3a+30

, taking away

3a

from both sides leaves unknown groups on the left. Then undo the added numbers to find

a

.

Use whole-number calculations carefully. If a step gives a whole number, it may be a solution, but it still needs a check. A check shows whether the number makes the original equation true.

a=b\Rightarrow a-c=b-c

Undo addition by taking away the same amount on both sides.
Undo equal groups by splitting into that many groups.
If unknown terms appear on both sides, remove matching groups from both sides.

Verify by substitution

To verify means to check that an answer is correct. Substitution means replacing the letter in the original equation with the number you found.

Work out both sides using the usual order: do multiplication before addition or subtraction. Then compare the totals. If the totals are equal, the answer works. If they are not equal, go back and look for an error.

Always check in the original equation, not only in the last line of your solving. The original equation is the one you were asked to solve.

\text{left side}=?\text{right side}

Put the proposed number in place of the letter.
Calculate each side and compare the results.
Equal totals confirm the solution.

Plan your work

Read the equation and identify its unknown and its terms. Decide what operation is being done to the unknown. Undo one part at a time, doing the same thing to both sides. Write each new equation clearly.

After finding a value, put it into the original equation and calculate both sides. This final check is part of solving, not an extra step.

Name the unknown.
Keep the two sides equal at every step.
Verify the result in the original equation.

Worked example

Undo addition, then equal groups

Solve and verify

4n+18=70

.

Remove the extra 18
The left side has 18 added to the groups of $n$ . Take away 18 from both sides so the equality stays true.
$4n=52$
Split into four equal groups
Four equal groups make 52. Divide both sides by 4 to find the amount in one group.
$n=13$
Check in the original equation
Replace $n$ with 13. Both sides have the same total, so 13 is a solution.
$4\times13+18=70$

Answer:

n=13

Check:

52+18=70

Worked example

Unknown groups on both sides

Solve and verify

5a+12=3a+30

.

Remove three groups from each side
Both sides have at least three groups of $a$ . Take away three groups from each side. The left side then has two groups, and the equality remains true.
$2a+12=30$
Remove 12, then split
Take away 12 from both sides. The two groups now total 18, so split 18 into two equal groups.
$2a=18,\quad a=9$
Check in the original equation
Put 9 in place of $a$ . Both sides equal 57, so the solution is correct.
$5\times9+12=3\times9+30$

Answer:

a=9

Check:

57=57

Worked example

Work through two added amounts

Solve and verify

6m+14=2m+46

.

Keep the remaining groups
Take away two groups of $m$ from both sides. This leaves four groups on the left and none on the right.
$4m+14=46$
Find one group
Take away 14 from both sides. Then split 32 into four equal groups.
$4m=32,\quad m=8$
Verify
Substitute 8 into the original equation and calculate each side. Both sides equal 62.
$6\times8+14=2\times8+46$

Answer:

m=8

Check:

62=62

Common mistakes and how to avoid them

Taking away a number from only one side.

Correction: Do the same operation to both sides so they remain equal.

Treating

4n

as

4+n

.

Correction:

4n

means four equal groups of

n

, or

4

multiplied by

n

.

Stopping after finding a possible value.

Correction: Substitute the value into the original equation and compare both totals.

Lesson summary

An equation states that two amounts are equal.
Undo operations while doing the same thing to both sides.
Verify by putting the answer into the original equation and checking both sides.

Check your understanding

Question 1

Solve

3x+8=29

. Which value makes the equation true?

$x=5$
$x=7$
$x=9$
correctIndex

Show answer and explanation

x=7

Take away 8 to get

3x=21

. Split 21 into three equal groups:

x=7

. Checking gives

3\times7+8=29

.

Question 2

Solve

4p+10=2p+24

. What is

p

?

$p=5$
$p=7$
$p=17$
correctIndex

Show answer and explanation

p=7

Take away

2p

from both sides to get

2p+10=24

. Take away 10, then split 14 into two groups. So

p=7

.

Key terms

Equation: A number sentence that says two amounts are equal.
Term: One part of an expression, separated from another part by a plus or minus sign.
Unknown: A number that a letter stands for when we are solving an equation.
Verify: Check that a proposed answer makes the original equation true.
Substitution: Replacing a letter in an equation with a number.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation C2.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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C2.3 · Solve and verify multi-term whole-number equations

Ontario Grade 6 Mathematics

Use equal amounts on both sides to find an unknown and check your answer.

What you will learn

Start with equal amounts

Undo operations while keeping equality

Verify by substitution

Plan your work

Undo addition, then equal groups

Unknown groups on both sides

Work through two added amounts

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Grade 6 Math

About this lesson and its review