G2 Solve first-degree equations including fractional coefficients

Ontario Grade 10 Mathematics

Modelling Linear Relations

Keep both sides balanced as you isolate the variable.

An equation says that two expressions have the same value. Solving it means finding the value of the variable that makes the statement true. In this lesson, you will use familiar operations to isolate the variable, including when its coefficient is a fraction. An equation stays balanced when you do the same operation to both sides.

What you will learn

Recognize a first-degree equation and identify its variable and coefficients.
Solve equations by using inverse operations on both sides.
Clear fractional coefficients by multiplying every term by a suitable common denominator.
Check a solution by substituting it into the original equation.

1. Grade 9 bridge: keep an equation balanced

You may already know how to solve an equation such as

x+5=12

. The variable

x

stands for a number you do not yet know. Subtracting

5

from both sides gives

x=7

. The equality remains true because both sides changed by the same amount.

A first-degree equation is an equation in which the variable has an exponent of

1

. For example,

3x-4=11

is first-degree. The number multiplying the variable is its coefficient. In

3x

, the coefficient is

3

.

The goal is to isolate the variable: get it by itself on one side. Undo addition or subtraction first, then undo multiplication or division. These are inverse operations because one reverses the other.

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

Do the same operation to both sides of an equation.
Use inverse operations to isolate the variable.
A fractional coefficient is a fraction multiplying the variable, such as $\frac{2}{3}x$ .

2. Solve with a fractional coefficient

A fraction in an equation does not change the balance rule. You can divide by a fractional coefficient, but multiplying every term by a common denominator often makes the arithmetic easier.

A denominator is the bottom number in a fraction. A common denominator is a number that each denominator divides evenly. Multiplying every term on both sides by that number clears the fractions. It does not change the solution, because the same non-zero factor is applied to both sides.

Be careful to multiply every term, not just the terms that contain a fraction. If a side has a sum or difference, each part of that side is a term. Brackets can help show which terms must be multiplied.

\frac{a}{d}=\frac{b}{d}\Rightarrow a=b (d\ne0)

Choose a common denominator for the fractions in the equation.
Multiply every term on both sides by that denominator.
Simplify, then use inverse operations to isolate the variable.
Substitute the answer into the original equation to check it.

3. A visual way to track the steps

Think of an equation as two equal piles. Multiplying both piles by the same number keeps them equal. For example, if each pile is multiplied by

6

, each individual term on both sides must be multiplied by

6

. That is why a constant term cannot be left unchanged.

Here is the structure for clearing thirds and halves. The common denominator is

6

, since both

3

and

2

divide evenly into

6

. The table shows how each kind of term changes. Use this idea before simplifying a longer equation.

The multiplier applies to every term on both sides.
After clearing denominators, solve the resulting equation as usual.
A check uses the original equation, so it can catch errors made while clearing fractions.

4. Guided example and independent practice

In the worked example, the equation includes both a fractional coefficient and a whole-number term. Multiplying every term by the denominator removes the fraction. The remaining steps use only addition, subtraction, multiplication, or division on both sides.

For independent practice, try

\frac{x}{4}+3=8

and

\frac{2x}{3}-1=5

. First choose a denominator that clears any fractions. Then solve and check by substitution. The answers are

x=20

for the first equation and

x=9

for the second.

Write one clear operation at a time.
When checking, evaluate both sides separately and compare their values.
If both sides match, the value satisfies the equation.

Multiplying each term by the common denominator

Original term	After multiplying by $6$
$\frac{x}{3}$	$2x$
$\frac{x}{2}$	$3x$
$4$	$24$

Worked example

Clear the fraction, then isolate the variable

Solve

\frac{2}{3}x+5=13

.

Find a common denominator
The only denominator is $3$ . Multiply every term on both sides by $3$ . The whole-number terms must also be multiplied.
$3(\frac{2}{3}x+5)=3(13)$
Simplify
The factors of $3$ and the denominator $3$ cancel in the fraction. Multiply the other terms to get an equation without fractions.
$2x+15=39$
Undo addition
Subtract $15$ from both sides to leave the variable term by itself on the left.
$2x=24$
Undo multiplication
Divide both sides by $2$ , the coefficient of $x$ .
$x=12$
Check in the original equation
Substitute $12$ for $x$ . The left side becomes $\frac{2}{3}(12)+5=8+5=13$ , which matches the right side.
$\frac{2}{3}(12)+5=13$

Answer:

x=12

.

Check: Substitution makes both sides equal to

13

.

Common mistakes and how to avoid them

Multiplying only the fraction by the common denominator.

Correction: Multiply every term on both sides. For example, multiplying

\frac{x}{3}+4

by

6

gives

2x+24

.

Changing a sign while moving a term without showing the operation.

Correction: Keep the equation balanced by writing the same addition or subtraction on both sides.

Stopping after finding a value without checking it.

Correction: Substitute the value into the original equation and confirm that both sides have the same value.

Lesson summary

An equation remains balanced when the same operation is applied to both sides.
Isolate the variable using inverse operations.
To clear fractional coefficients, multiply every term on both sides by a common denominator.
Check a solution by substituting it into the original equation.

Check your understanding

Question 1

Solve

\frac{x}{5}+2=6

.

$x=20$
$x=4$
$x=8$
$x=30$

Show answer and explanation

x=20

Subtract

2

from both sides to get

\frac{x}{5}=4

. Multiply both sides by

5

, giving

x=20

.

Question 2

What is the result of multiplying every term in

\frac{x}{4}+3=7

by

4

?

$x+3=28$
$x+12=28$
$x+12=7$
$4x+12=28$

Show answer and explanation

x+12=28

Multiplying

\frac{x}{4}

by

4

gives

x

. The constant

3

also gets multiplied by

4

, giving

12

, and the right side becomes

28

.

Key terms

Equation: A statement that two expressions have the same value.
Variable: A letter that represents a number.
Coefficient: The number multiplying a variable.
First-degree equation: An equation in which the variable has exponent $1$ .
Isolate: Get a variable by itself on one side of an equation.
Common denominator: A number that each denominator divides evenly into.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G2. It is a study resource, not an official curriculum publication.

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G2 · Solve first-degree equations including fractional coefficients