MathematicsChapter 5: Linear Equations and Inequalities
Equivalent Equations
Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra
Learning objectives
- Explain equivalent equations in clear mathematical language
- Apply the ideas in add or subtract
- Solve problems involving multiply or divide
Definition
Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in. You keep an equation equivalent by adding or subtracting the same number on both sides, or by multiplying or dividing both sides by the same nonzero number. That is how 5x-8=12 becomes x=4, and how 3(x-2)=15 becomes x=7.
Mathematical form
x=7. Worked
x=4. Worked
x=7. Check by substituting back
Where / symbols
- equivalent equations
- Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
- same solution set
- Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
- 5x-8=12
- That is how 5x-8=12 becomes x=4, and how 3(x-2)=15 becomes x=7.
- x=4
- That is how 5x-8=12 becomes x=4, and how 3(x-2)=15 becomes x=7.
Relationship
- 5x-8=12
- x=4
- 3(x-2)=15
- x=7
Overview
Two legal moves that do not change the root:
Add Or Subtract
Adding or subtracting the same number on both sides keeps the balance. From 5x-8=12 add 8 to both sides to get 5x=20. The solution set is unchanged; you have only peeled the constant away from the unknown. The same idea undoes a subtraction on the other side of an equation.
Examples 5x-8=12 becomes 5x=20 · x+3=10 becomes x=7 · 2x-5=9 becomes 2x=14
Multiply Or Divide
Multiply or divide both sides by the same nonzero number. From 5x=20 divide by 5 to get x=4. From 3(x-2)=15 divide both sides by 3 to get x-2=5, then add 2 to get x=7. Never divide by zero: that move is illegal and the new equation is not equivalent.
Examples 5x=20 so x=4 · 3(x-2)=15 so x=7 · 4x=18 so x=9/2
What it means
Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
Main ideas
1. ADD OR SUBTRACT: Adding or subtracting the same number on both sides keeps the balance. 2. MULTIPLY OR DIVIDE: Multiply or divide both sides by the same nonzero number.
Why it matters
Worked: 5x-8=12 => 5x=20 => x=4. Worked: 3(x-2)=15 => x-2=5 => x=7. Check by substituting back: 5(4)-8=12 and 3(7-2)=15.
How to answer in an exam
Equivalent equations share the same solution set, not merely a similar look. Multiplying both sides by zero, or dividing by an expression that might be zero, can destroy equivalence. Always check the root in the original equation.
Worked example
- 5x-8=12 => 5x=20 => x=4. Worked: 3(x-2)=15 => x-2=5 => x=7. Check by substituting back: 5(4)-8=12 and 3(7-2)=15.
In simple words
Equivalent equations share the same solution set, not merely a similar look. Multiplying both sides by zero, or dividing by an expression that might be zero, can destroy equivalence. Always check the root in the original equation.
Key points
- equivalent equations
- same solution set
- 5x-8=12
- x=4
- same number on both sides
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Equivalent Equations
Sticky Notes · Quick Revision
Core Idea
- Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
Key Terms
- equivalent equations: Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
- same solution set: Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
Main Pattern
- 1. ADD OR SUBTRACT: Adding or subtracting the same number on both sides keeps the balance.
- 2. MULTIPLY OR DIVIDE: Multiply or divide both sides by the same nonzero number.
Worked Reminder
- 5x-8=12 becomes 5x=20
- x+3=10 becomes x=7
- 2x-5=9 becomes 2x=14
Common Mistake
- Equivalent equations share the same solution set, not merely a similar look. Multiplying both sides by zero, or dividing by an expression that might be zero, can destroy equivalence. Always check the root in the original equation.
Exam Tip
- State the rule for equivalent equations, show each algebraic step, and check the final answer against the question.
60-second revision
- Two equations are equivalent equations when they have the same solution set: every number that works in one works in the other, and no extra roots sneak in.
- Adding or subtracting the same number on both sides keeps the balance.
- Multiply or divide both sides by the same nonzero number.
- Worked: 5x-8=12 => 5x=20 => x=4.
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