Linear Equation in One Variable

Linear Equation in One Variable

What is it?

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.

In simple words

• 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.

Important 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.

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.

What you should learn

• 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

Examples

• 5x-8=12
• 3(x-2)=15
• (12)x + 4 = 7

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.

WHAT IS NOT LINEAR

If the unknown is squared, or two unknowns are multiplied, the equation is not linear in one variable. x^2=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.

Full school-level explanation

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.

Check the power of the unknown and how many letters you have:

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.

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.

Remember

• ax+b=0
• a!=0
• highest power of x is 1

Examples

• 5x-8=12
• 3(x-2)=15
• (12)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. x^2=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.

Remember

• x²=9
• xy=12
• not linear

Examples

• x²=9
• xy=12
• x³=8

Remember

• ax+b=0
• a!=0
• exponent 1
• exactly one solution
• ax+b=0
• a!=0
• exponent 1
• exponent 1

How the ideas connect

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.

Deeper understanding

Go beyond memorising Linear Equation in One Variable. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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.

Common mistakes

Exam focus

Core Idea

Key Terms

Main Pattern

Worked Reminder

Exam focus

Quick revision

• ax+b=0
• a!=0
• exponent 1
• exactly one solution
• 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.

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.

For long answers on Linear Equation in One Variable: define clearly → explain the main parts → give one example → state one exam caution.