MathematicsChapter 5: Linear Equations and Inequalities
Linear Equation in One Variable
A balanced scale is a picture of an equation: both sides have the same value.
Learning objectives
- Explain linear equation in one variable in clear mathematical language
- Apply the ideas in what counts as linear
- Solve problems involving what is not linear
Definition
A balanced scale is a picture of an equation: both sides have the same value. A linear equation in one variable can be written as ax+b=0, where a and b are real numbers and a!=0. The unknown has exponent 1, so you are not dealing with squares, cubes, or products of two different letters. Because the power is 1 and a is not zero, there is exactly one solution. That unique value is the root of the equation.
Formula
b=0 and look at a. If a
Where / symbols
- ax+b=0
- A linear equation in one variable can be written as ax+b=0, where a and b are real numbers and a!=0.
- a!=0
- A linear equation in one variable can be written as ax+b=0, where a and b are real numbers and a!=0.
- exponent 1
- The unknown has exponent 1, so you are not dealing with squares, cubes, or products of two different letters.
- exactly one solution
- Because the power is 1 and a is not zero, there is exactly one solution.
Relationship
- ax+b=0
- a!=0
Overview
Check the power of the unknown and how many letters you have:
What Counts As Linear
After you expand brackets and collect like terms, a linear equation in x looks like ax+b=0 with a!=0. You may start from 5x-8=12 or 3(x-2)=15; those are still linear because the highest power of x is 1. A constant on the other side does not spoil the form: move it across and you still have one unknown to the first power.
Examples 5x-8=12 · 3(x-2)=15 · (1/2)x + 4 = 7
What Is Not Linear
If the unknown is squared, or two unknowns are multiplied, the equation is not linear in one variable. x2=9 has two roots, +3 and -3, so it already fails exactly one solution. xy=12 has two letters multiplied, so it is not an equation in a single unknown of degree 1. Cubes such as x^3=8 are also not linear.
Examples x^2=9 · xy=12 · x^3=8
What it means
A balanced scale is a picture of an equation: both sides have the same value.
Main ideas
1. WHAT COUNTS AS LINEAR: After you expand brackets and collect like terms, a linear equation in x looks like ax+b=0 with a!=0. 2. WHAT IS NOT LINEAR: If the unknown is squared, or two unknowns are multiplied, the equation is not linear in one variable.
Why it matters
Rewrite in the shape ax+b=0 and look at a. If a=0 you have either a true statement for every x or a false statement for every x, not a linear equation to solve. Keep a!=0 and exponent 1.
How to answer in an exam
Do not call x^2=9 linear just because it looks short. Linear in one variable means exponent 1, a!=0, and exactly one solution. Products such as xy=12 belong to a later chapter on two variables.
Explanation
Rewrite in the shape ax+b=0 and look at a. If a=0 you have either a true statement for every x or a false statement for every x, not a linear equation to solve. Keep a!=0 and exponent 1.
In simple words
Do not call x^2=9 linear just because it looks short. Linear in one variable means exponent 1, a!=0, and exactly one solution. Products such as xy=12 belong to a later chapter on two variables.
Key points
- ax+b=0
- a!=0
- exponent 1
- exactly one solution
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Linear Equation in One Variable
Sticky Notes · Quick Revision
Core Idea
- A balanced scale is a picture of an equation: both sides have the same value.
Key Terms
- ax+b=0: A linear equation in one variable can be written as ax+b=0, where a and b are real numbers and a!=0.
- a!=0: A linear equation in one variable can be written as ax+b=0, where a and b are real numbers and a!=0.
Main Pattern
- 1. WHAT COUNTS AS LINEAR: After you expand brackets and collect like terms, a linear equation in x looks like ax+b=0 with a!=0.
- 2. WHAT IS NOT LINEAR: If the unknown is squared, or two unknowns are multiplied, the equation is not linear in one variable.
Worked Reminder
- 5x-8=12
- 3(x-2)=15
- (12)x + 4 = 7
Common Mistake
- Do not call x^2=9 linear just because it looks short. Linear in one variable means exponent 1, a!=0, and exactly one solution. Products such as xy=12 belong to a later chapter on two variables.
Exam Tip
- State the rule for linear equation in one variable, show each algebraic step, and check the final answer against the question.
60-second revision
- A balanced scale is a picture of an equation: both sides have the same value.
- After you expand brackets and collect like terms, a linear equation in x looks like ax+b=0 with a!=0.
- If the unknown is squared, or two unknowns are multiplied, the equation is not linear in one variable.
- Rewrite in the shape ax+b=0 and look at a.
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