Section A — Chapters 1–22
Solving Linear Equations
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to understand what a linear equation is.
- ✓By the end of this lesson students will be able to solve simple linear equations involving addition, subtraction, multiplication, and division.
- ✓By the end of this lesson students will be able to solve linear equations with constants and/or variables on both sides.
- ✓By the end of this lesson students will be able to solve linear equations involving brackets.
- ✓By the end of this lesson students will be able to check their solutions by substitution.
Key concepts
An equation where the highest power of the variable (usually 'x') is 1. It can be written in the general form ax + b = 0, where 'a' and 'b' are constants (numbers) and 'x' is the variable.
The process of finding the value(s) of the variable that make the equation true. This value is called the solution or root of the equation.
The fundamental principle for solving equations. Whatever mathematical operation (addition, subtraction, multiplication, division) you perform on one side of the equation, you must perform the exact same operation on the other side to keep the equation balanced and true. This is often referred to as 'doing the same to both sides'.
Equations that require one or two inverse operations to isolate the variable. Examples include x + 7 = 15 or 4x = 20.
Equations where numerical values (constants) appear on both the left-hand side and the right-hand side of the equals sign, in addition to the variable term. For example, x + 3 = 10 - 2.
Equations where the variable appears on both the left-hand side and the right-hand side of the equals sign. For example, 5x = 2x + 9.
Equations that combine both variable terms and constant terms on both sides of the equals sign. For example, 4x + 7 = 2x + 15.
Equations that contain terms enclosed in brackets. The first step to solving these is usually to expand the brackets using the distributive law (multiply the term outside the bracket by each term inside). For example, 3(x - 2) = 12.
Key facts to remember
- 1The primary goal when solving a linear equation is to isolate the variable (e.g., x) on one side of the equals sign.
- 2To maintain the balance of an equation, any operation performed on one side must also be performed on the other side.
- 3Use inverse operations: addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication.
- 4When an equation contains brackets, expand them first using the distributive law.
- 5It is generally good practice to gather all variable terms on one side and all constant terms on the other side.
- 6Always check your solution by substituting the found value of the variable back into the original equation to ensure both sides are equal.
Worked examples
Example 1
Solve the equation: 3x + 7 = 22 - 3
Answer
x = 4
Always simplify each side of the equation as much as possible before moving terms across the equals sign.
Example 2
Solve the equation: 6x - 5 = 2x + 11
Answer
x = 4
It's often easier to move the smaller variable term to the side with the larger variable term to avoid negative coefficients.
Example 3
Solve the equation: 4(x - 3) = 20
Answer
x = 8
Remember the distributive law: a(b + c) = ab + ac.
Common mistakes
- ✗Forgetting to perform an operation on both sides of the equation, leading to an unbalanced equation.
- ✗Making arithmetic errors, especially with positive and negative numbers, when adding, subtracting, multiplying, or dividing.
- ✗Incorrectly expanding brackets, particularly when there is a negative sign outside the bracket or negative terms inside.
- ✗Confusing inverse operations (e.g., adding 5 when you should subtract 5 to move a term).
- ✗Not simplifying each side of the equation fully before attempting to move terms across the equals sign.
Exam tips
- ★Show all your steps clearly. Even if your final answer is incorrect, you can still earn partial marks for correct method.
- ★Always check your answer by substituting it back into the original equation. This helps you catch mistakes and ensures your solution is correct.
- ★Be very careful with signs (positive and negative numbers) throughout your calculations, as a single sign error can lead to an incorrect answer.
- ★If you get stuck, try to simplify each side of the equation first, then focus on moving variable terms to one side and constant terms to the other.
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