Equivalent Equations

Equivalent Equations

What is it?

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.

In simple words

• 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.

Important 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.

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.

What you should learn

• Explain equivalent equations in clear mathematical language
• Apply the ideas in add or subtract
• Solve problems involving multiply or divide

Examples

• 5x-8=12 becomes 5x=20
• x+3=10 becomes x=7
• 2x-5=9 becomes 2x=14

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.

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.

Full school-level explanation

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.

Two legal moves that do not change the root:

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.

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.

Remember

• same number on both sides
• 5x-8=12
• 5x=20

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.

Remember

• nonzero number
• x=4
• x=7

Examples

• 5x=20 so x=4
• 3(x-2)=15 so x=7
• 4x=18 so x=92

Remember

• equivalent equations
• same solution set
• 5x-8=12
• x=4
• 3(x-2)=15
• x=7
• x=4
• x=7

How the ideas connect

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.

Deeper understanding

Go beyond memorising Equivalent Equations. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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.

Common mistakes

Exam focus

Core Idea

Key Terms

Main Pattern

Worked Reminder

Exam focus

Quick revision

• equivalent equations
• same solution set
• 5x-8=12
• x=4
• equivalent equations: Two equations are equivalent equations when they have the same solution set: every number that works
• same solution set: Two equations are equivalent equations when they have the same solution set: every number that works

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.

For long answers on Equivalent Equations: define clearly → explain the main parts → give one example → state one exam caution.