Isolate and evaluate a variable in a formula

Ontario Grade 10 Mathematics

Modelling Linear Relations

Use inverse operations to rearrange a formula, then substitute known values

A formula is a rule that connects quantities. For example, the formula for the perimeter of a rectangle uses its length and width. Sometimes you know the measurements and need to calculate a result. At other times, you need to rearrange the formula first so that a different variable is alone. You already know the key idea from solving equations: do the same operation to both sides. This lesson uses that idea with formulas.

What you will learn

Explain what it means to isolate a variable in a formula.
Use inverse operations to rearrange a formula while keeping both sides balanced.
Substitute known values into a formula and calculate the unknown value.
Check whether a result makes sense in the original formula.

1. Bridge: equations and balanced sides

An equation says that two expressions have the same value. The equals sign means “is equal to.” To keep an equation true, you must do the same operation to both sides.

An inverse operation undoes another operation. Addition and subtraction are inverses. Multiplication and division are inverses. For example, subtracting a number undoes adding that number.

A variable is a letter that stands for a number or measurement. To isolate a variable means to get it alone on one side of an equation. In a formula, the variable you want to find may not be alone at first.

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

Keep both sides balanced by applying the same operation to each side.
Use inverse operations to undo operations around the variable.
Choose the variable you want to isolate before you begin.

2. Rearrange a formula

A formula is an equation that describes a relationship between quantities. The perimeter formula for a rectangle is

P=2l+2w

. Here,

P

means perimeter,

l

means length, and

w

means width.

Suppose you know the perimeter and width but need the length. Start with the original formula and undo the operations attached to the length. First subtract

2w

from both sides. Then divide both sides by

2

. These steps leave

l

alone.

The order matters. Undo addition or subtraction before undoing multiplication or division when those operations are applied to the variable. Use the same operation on both sides at each step.

A rearranged formula is still the same relationship, written to make a different variable the subject. The subject is the variable that is alone on one side.

P=2l+2w \Rightarrow l=\frac{P-2w}{2}

Write the original formula before changing it.
Undo operations in a sensible reverse order.
Keep the equals sign and both sides balanced.

3. Evaluate a variable

To evaluate a variable means to calculate its value using known values in a formula. Substitution means replacing a variable with a given number.

Use parentheses around substituted negative numbers or values that contain more than one part. Follow the usual order of operations: calculate inside parentheses first, then multiply or divide, then add or subtract.

Include units when the quantities have units. Check that the final value is reasonable and, when possible, place it back into the original formula to confirm it works.

l=\frac{P-2w}{2}

Substitute each known value into the rearranged formula.
Calculate carefully using the order of operations.
Use the original formula to check the result.

4. A reliable routine

For a question that asks you to find a variable, first identify which variable is unknown. If it is not alone, rearrange the formula to isolate it. Then substitute the values you know and calculate.

For a question that asks you to rearrange a formula but gives no numbers, stop once the requested variable is isolated. Do not invent values.

A useful final check is to substitute your calculated value into the original formula. Both sides should have the same value. If they do not, review your rearranging steps and arithmetic.

A clear sequence is to write the original formula, isolate the target variable, substitute known values, and check the result.

Identify the target variable.
Isolate it using inverse operations.
Substitute known values and calculate.
Check the answer in the original formula.

Undoing operations to isolate the length

Stage	Equation	What changes
Original	$P=2l+2w$	Length is multiplied by 2, then $2w$ is added.
Subtract $2w$	$P-2w=2l$	Undo the addition on both sides.
Divide by 2	$\frac{P-2w}{2}=l$	Undo the multiplication on both sides.
Write with the target first	$l=\frac{P-2w}{2}$	The length is isolated.

Worked example

Find the length of a rectangle

A rectangle has a perimeter of 34 cm and a width of 6 cm. Use the perimeter formula to find its length.

Choose the variable
The unknown is the length, so we want $l$ alone. The given values are $P=34$ and $w=6$ .
Isolate the length
Start from the perimeter formula. Subtract $2w$ from both sides to undo the addition of $2w$ . Then divide both sides by $2$ to leave $l$ alone.
$P=2l+2w$
Substitute the known values
Replace $P$ with $34$ and $w$ with $6$ in the isolated formula. This uses the same relationship, now arranged to calculate length.
$l=\frac{34-2(6)}{2}$
Calculate and check
Multiply first, then subtract and divide. The result is $11$ cm. Check it in the original formula: twice the length plus twice the width gives the stated perimeter.
$l=11,\qquad 2(11)+2(6)=34$

Answer: The length is 11 cm.

Check: Substituting

l=11

cm and

w=6

cm into the original formula gives

2(11)+2(6)=34

cm, which matches the given perimeter.

Common mistakes and how to avoid them

Changing only one side when rearranging a formula.

Correction: Apply the same operation to both sides so the equation stays balanced.

Dividing by 2 before removing the added

2w

in

P=2l+2w

.

Correction: Subtract

2w

from both sides first, then divide by 2.

Substituting values into the original formula without isolating the requested variable.

Correction: If the requested variable is not alone, rearrange the formula first.

Stopping after finding a number without checking it.

Correction: Substitute the result into the original formula and confirm the relationship works.

Lesson summary

A formula is an equation connecting quantities.
To isolate a variable, use inverse operations and do the same thing to both sides.
To evaluate a variable, substitute known values and calculate in the correct order.
Check a calculated value in the original formula.

Check your understanding

Question 1

Rearrange

A=lw

to make

w

the subject.

$w=A-l$
$w=\frac{A}{l}$
$w=Al$
$w=\frac{l}{A}$

Show answer and explanation

w=\frac{A}{l}

Divide both sides by

l

to undo multiplication by

l

. This gives

w=A/l

.

Question 2

Use

d=vt

to find

t

when

d=48

km and

v=8

km/h.

6 h
40 h
56 h
384 h

Show answer and explanation

6 h

Divide distance by speed:

t=d/v=48/8=6

h.

Question 3

Which operation should be used first to isolate

x

in

y=3x+7

?

Add 7 to both sides
Subtract 7 from both sides
Multiply both sides by 3
Divide both sides by 7

Show answer and explanation

Subtract 7 from both sides

Subtract 7 from both sides to undo the addition. Then divide by 3 to isolate

x

.

Key terms

Formula: An equation that describes a relationship between quantities.
Variable: A letter that stands for a number or measurement.
Isolate: Get a chosen variable alone on one side of an equation.
Inverse operation: An operation that undoes another operation, such as subtraction undoing addition.
Substitution: Replacing a variable with a known value.
Evaluate: Calculate the value of a variable or expression using known values.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G3. 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.

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