Strand 4 — Algebra

Linear Equations and Inequalities

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to solve linear equations in one variable.
✓By the end of this lesson students will be able to solve linear inequalities in one variable and represent solutions on a number line.
✓By the end of this lesson students will be able to solve simultaneous linear equations in two variables (2x2 systems) using both substitution and elimination methods.
✓By the end of this lesson students will be able to apply linear equations and inequalities to solve real-world problems.

Key concepts

Linear Equation (One Variable)

A linear equation in one variable is an algebraic equation where the highest power of the variable is 1. It typically has one unique solution.

ax + b = c

Solving Linear Equations

To solve a linear equation, we aim to isolate the variable on one side of the equals sign. This involves performing inverse operations (addition/subtraction, multiplication/division) to both sides of the equation to maintain balance.

Linear Inequality (One Variable)

A linear inequality in one variable is a mathematical statement that compares two expressions using an inequality symbol: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, inequalities typically have a range of solutions.

ax + b < c (or >, ≤, ≥)

Solving Linear Inequalities

The process is similar to solving linear equations, but with one crucial difference: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

Simultaneous Linear Equations (2x2)

A system of two linear equations with two variables (e.g., x and y) is called simultaneous linear equations. The goal is to find the values for both variables that satisfy both equations at the same time. The solution is an ordered pair (x, y).

a₁x + b₁y = c₁ a₂x + b₂y = c₂

Substitution Method

This method involves solving one of the equations for one variable in terms of the other, and then substituting that expression into the second equation. This reduces the system to a single equation with one variable.

Elimination Method

This method involves manipulating one or both equations (by multiplying by constants) so that when the equations are added or subtracted, one of the variables is eliminated. This leaves a single equation with one variable.

Key facts to remember

1To solve an equation, perform the same operation on both sides to maintain balance.
2When solving inequalities, reverse the inequality sign if multiplying or dividing by a negative number.
3A solution to a linear equation in one variable is a single value.
4A solution to a linear inequality in one variable is a range of values, often represented on a number line.
5Simultaneous equations can be solved using the substitution method or the elimination method.
6The solution to a system of 2x2 linear equations is an ordered pair (x, y).
7Always check your solutions by substituting them back into the original equations/inequalities.

Worked examples

Example 1

Solve the equation: 3(x - 2) + 5 = 2x + 7

I3(x - 2) + 5 = 2x + 7

II3x - 6 + 5 = 2x + 7 (Distribute the 3)

III3x - 1 = 2x + 7 (Combine like terms on the left side)

IV3x - 2x - 1 = 7 (Subtract 2x from both sides)

Vx - 1 = 7

VIx = 7 + 1 (Add 1 to both sides)

VIIx = 8

Answer

x = 8

Always check your answer by substituting it back into the original equation.

Example 2

Solve the inequality: 5 - 2x ≤ 11, and represent the solution on a number line.

I5 - 2x ≤ 11

II-2x ≤ 11 - 5 (Subtract 5 from both sides)

III-2x ≤ 6

IVx ≥ 6 / (-2) (Divide by -2 and reverse the inequality sign)

Vx ≥ -3

VITo represent on a number line: Draw a number line. Place a closed (filled-in) circle at -3, and draw an arrow extending to the right, indicating all numbers greater than or equal to -3.

Answer

x ≥ -3

Remember to reverse the inequality sign when multiplying or dividing by a negative number.

Example 3

Solve the following simultaneous equations using the elimination method: 1. 2x + y = 8 2. 3x - 2y = 5

IGiven equations:

II(1) 2x + y = 8

III(2) 3x - 2y = 5

IVStep 1: Multiply equation (1) by 2 to make the coefficients of 'y' opposites:

V2 * (2x + y) = 2 * 8

VI(3) 4x + 2y = 16

VIIStep 2: Add equation (3) and equation (2) to eliminate 'y':

VIII 4x + 2y = 16

9+ 3x - 2y = 5

10----------------

11 7x = 21

12Step 3: Solve for 'x':

137x = 21

14x = 21 / 7

15x = 3

16Step 4: Substitute the value of 'x' (x=3) into one of the original equations (e.g., equation 1) to find 'y':

172(3) + y = 8

186 + y = 8

19y = 8 - 6

20y = 2

21Step 5: Check the solution (x=3, y=2) in the other original equation (equation 2):

223(3) - 2(2) = 5

239 - 4 = 5

245 = 5 (The solution is correct)

Answer

x = 3, y = 2

The substitution method could also be used. For example, from (1) y = 8 - 2x, then substitute this into (2).

Common mistakes

✗Incorrectly applying the distributive property, e.g., 3(x - 2) becoming 3x - 2 instead of 3x - 6.
✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
✗Making arithmetic errors, especially with negative numbers, when combining like terms or transposing terms.
✗Not checking solutions, particularly for simultaneous equations, which can lead to errors going unnoticed.
✗Mixing up the signs when adding or subtracting equations in the elimination method, or making errors when multiplying equations.

Exam tips

★Always show all steps clearly, even for simple calculations, as partial credit is often awarded for correct methods.
★Double-check your arithmetic, especially when dealing with negative numbers, as small errors can lead to incorrect final answers.
★For inequalities, remember to draw the number line if specifically asked, using open/closed circles correctly.
★When solving simultaneous equations, choose the method (substitution or elimination) that seems most efficient for the given equations to save time.
★Verify your solution by substituting the values back into the *original* equations to ensure they satisfy all conditions.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum