Sec 1 Maths Chapter 5

Linear Equations

Learn how to solve linear equations, handle fractional coefficients, translate word problems into equations and use mathematical formulae.

This page teaches the key ideas and checks for common equation-solving mistakes. Use the PDF for the full printable classwork and homework practice.

Download PDF Take Quick Self-Check Preview practice

Linear equations infographic showing balance scale inverse operations fractional equations word problems and formulae for Sec 1 Maths

Learning goals

What this chapter trains

Linear equations teach students to keep both sides balanced, isolate the unknown and check whether a solution satisfies the original equation.

Understand equationsKnow the difference between an expression, an equation and a linear equation.

Solve linear equationsUse inverse operations while keeping both sides equal.

Handle fractional equationsClear denominators carefully and watch for restrictions.

Apply equationsTranslate word problems and formulae into equation-solving steps.

Page map

1Concept of equation 2Solving linear equations 3Fractional coefficients and equations 4Applications in real-world contexts 5Mathematical formulae 6Common equation traps 7Quick Self-Check 8Practice preview

Concept 1

Concept of equation

An equation is a mathematical statement that says two expressions have equal value. The equals sign means both sides must balance.

Expression

An expression has no equals sign.

\[ 3x + 5 \]

Equation

An equation states that two expressions are equal.

\[ x + 5 = 12 \]

Solution

A solution is a value that makes the equation true.

\[ x = 7 \]

Exam trap Do not call an expression an equation. An equation must have an equals sign.

Concept 2

Solving linear equations

To solve an equation, isolate the unknown on one side. Whatever operation is done to one side must also be done to the other side.

Balance idea

If both sides are equal, adding, subtracting, multiplying or dividing both sides by the same non-zero number keeps the equation balanced.

\[ \begin{aligned} x + 5 &= 12 \\ x + 5 – 5 &= 12 – 5 \\ x &= 7 \end{aligned} \]

Worked example

Solve $3x – 4 = 11$.

Reveal teaching answer

\[ \begin{aligned} 3x – 4 &= 11 \\ 3x &= 15 \\ x &= 5 \end{aligned} \]

Check: $3(5)-4=11$.

Concept 3

Fractional coefficients and fractional equations

Equations with fractional coefficients can be solved by clearing denominators. Fractional equations require extra care because the denominator cannot be zero.

Fractional coefficient

Solve $\frac{x}{2}+3=9$.

Reveal teaching answer

\[ \begin{aligned} \frac{x}{2}+3 &= 9 \\ \frac{x}{2} &= 6 \\ x &= 12 \end{aligned} \]

Fractional equation

Solve $\frac{9}{2x-5}=3$.

Reveal teaching answer

\[ \begin{aligned} \frac{9}{2x-5} &= 3 \\ 9 &= 3(2x-5) \\ 9 &= 6x – 15 \\ 24 &= 6x \\ x &= 4 \end{aligned} \]

Restriction: $2x-5 \ne 0$, so $x \ne \frac{5}{2}$. The solution $x=4$ is allowed.

Exam trap For fractional equations, check that the final answer does not make any denominator equal to zero.

Concept 4

Applications of linear equations

Algebra helps solve word problems that are hard to draw using models. The key skill is translating the situation into an equation.

Word problem flow

Three identical pens and a $2 delivery fee cost $11. Let $p$ be the cost of one pen.

\[ \begin{aligned} 3p + 2 &= 11 \\ 3p &= 9 \\ p &= 3 \end{aligned} \]

Age problem structure

A man is 3 times as old as his son. In 10 years, their ages add up to 76.

\[ \begin{aligned} x+10 + 3x+10 &= 76 \\ 4x + 20 &= 76 \\ x &= 14 \end{aligned} \]

The son is 14 years old now.

Concept 5

Mathematical formulae

A formula shows the relationship between quantities. Students may need to substitute values into a formula or construct a formula from a situation.

Evaluating formulae

Use $V=lbh$ to find $V$ when $l=7$, $b=4$, $h=3$.

Reveal teaching answer

\[ \begin{aligned} V &= lbh \\ V &= 7 \times 4 \times 3 \\ V &= 84 \end{aligned} \]

Constructing formulae

The sum of four consecutive even numbers can be represented using $n$, $n+2$, $n+4$, $n+6$.

\[ S=n+(n+2)+(n+4)+(n+6) \]

Trap-focused review

Common equation traps that cost marks

These are the mistakes students should check before starting the full PDF practice.

Changing one side only

If you subtract 5 from the left side, subtract 5 from the right side too.

Sign errors

When moving terms, think of applying inverse operations, not simply “moving across”.

Fractional equations

Clear denominators carefully and check restrictions such as $2x-1 \ne 0$.

Wrong word-problem variable

Define the unknown clearly before forming an equation.

Formula substitution

Substitute values into the correct variables. Do not mix up length, breadth and height.

Not checking answers

Substitute the answer back into the original equation, especially for fractional equations.

Readiness diagnostic

Quick Self-Check before PDF practice

This short quiz checks whether students are ready for the full Chapter 5 PDF. It focuses on equation traps, not practice volume.

Suggested classroom use: students attempt the quiz first. If they score low, review the trap cards before starting the PDF.

1. Solve $x+5=12$.

$x=5$ $x=7$ $x=17$

2. Solve $3x-4=11$.

$x=3$ $x=4$ $x=5$

3. Solve $\frac{x}{2}+3=9$.

$x=12$ $x=6$ $x=3$

4. In $\frac{5}{2x+1}$, what restriction must be stated?

$x \ne 1$ $x \ne 0$ $x \ne -\frac{1}{2}$

5. Use $V=lbh$. Find $V$ when $l=7$, $b=4$, $h=3$.

Practice preview

Diagnostic practice questions

These questions are for projection and discussion. Answers are hidden so students can attempt first. Use the PDF for the full classwork and homework set.

Solve $x-4=9$.

Skill tested: inverse operation.

Reveal diagnostic answer

\[ \begin{aligned} x-4 &= 9 \\ x &= 13 \end{aligned} \]

Solve $2x+3=9$.

Skill tested: two-step equation.

Reveal diagnostic answer

\[ \begin{aligned} 2x+3 &= 9 \\ 2x &= 6 \\ x &= 3 \end{aligned} \]

Solve $3x-4=11$.

Skill tested: balancing both sides.

Reveal diagnostic answer

\[ \begin{aligned} 3x-4 &= 11 \\ 3x &= 15 \\ x &= 5 \end{aligned} \]

Solve $\frac{x}{3}-2=4$.

Skill tested: fractional coefficient.

Reveal diagnostic answer

\[ \begin{aligned} \frac{x}{3}-2 &= 4 \\ \frac{x}{3} &= 6 \\ x &= 18 \end{aligned} \]

Solve $\frac{9}{2x-5}=3$.

Skill tested: fractional equation and restriction.

Reveal diagnostic answer

\[ \begin{aligned} \frac{9}{2x-5} &= 3 \\ 9 &= 3(2x-5) \\ 9 &= 6x-15 \\ x &= 4 \end{aligned} \]

Restriction: $x \ne \frac{5}{2}$, so $x=4$ is valid.

Three identical pens and a $2 delivery fee cost $11. Form and solve an equation.

Skill tested: word problem to equation.

Reveal diagnostic answer

Let $p$ be the cost of one pen.

\[ \begin{aligned} 3p+2 &= 11 \\ 3p &= 9 \\ p &= 3 \end{aligned} \]

Use $A=\frac{1}{2}bh$ to find $A$ when $b=12$, $h=9$.

Skill tested: substitution into formula.

Reveal diagnostic answer

\[ \begin{aligned} A &= \frac{1}{2}bh \\ A &= \frac{1}{2}(12)(9) \\ A &= 54 \end{aligned} \]

Find a formula for the sum $S$ of four consecutive odd numbers, starting with $n$.

Skill tested: constructing formulae.

Reveal diagnostic answer

\[ S=n+(n+2)+(n+4)+(n+6) \]

This can also be simplified to $S=4n+12$.

Download full Chapter 5 PDF Take Quick Self-Check

PDF-first practice

Use the PDF for classwork and homework

The webpage is designed for teaching, diagnosis and discussion. The PDF is the main practice asset for students.

Full chapter practice

The PDF covers solving equations, fractional equations, word problems and mathematical formulae.

LaTeX maths

Fractions and equations are typeset properly for a clearer worksheet experience.

Teacher-friendly

Use the page to teach and project. Use the PDF for student attempts and follow-up marking.

Download Chapter 5 PDF

Teacher notes

How to use this page in class

Suggested lesson flow

Teach the concept of balanceStart with $x+5=12$ and emphasise doing the same operation to both sides.

Move to fractional equationsUse display equations and restrictions before students attempt the PDF.

Run the Quick Self-CheckUse the score to decide whether students are ready for the full worksheet.

Use the PDF for practiceThe PDF carries the full classwork and homework load.

What to watch for

Students may apply an operation to only one side of the equation.

Students may lose negative signs when simplifying.

Students may forget restrictions in fractional equations.

Students may form the wrong equation from a word problem.

Students may substitute into the wrong variable in a formula.

Ready for Chapter 5 practice?

Download the PDF for the full worksheet, then use the self-check and diagnostic reveals when you need help with equation-solving traps.

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