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Teacher Guide: Equations with Variables on Both Sides

Learn how to solve equations where the variable appears on both sides of the equals sign.

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Worksheet Answer key

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10 questions on Equations. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Solve linear equations with variables on both sides
  • Apply the distributive property before solving when necessary
  • Recognize and handle identity equations (infinite solutions) and contradictions (no solution)
  • Check solutions by substitution
Prerequisites
  • • Solving one-step and two-step equations
  • • Combining like terms
  • • Distributive property
  • • Working with negative numbers
Discussion Starters
  • 1. Why do we need to get all the variables on one side?
  • 2. What real-life situation could be modeled by 5x+10=3x+305x + 10 = 3x + 30?
  • 3. Is x=5x = 5 the only solution to 2x+3=x+82x + 3 = x + 8? How do you know?
  • 4. What does it mean when an equation has no solution? Can you create an example?
Common Misconceptions

Moving a term to the other side without changing its sign

Remediation: Emphasize that 'moving' is actually subtracting (or adding). Write out both sides explicitly: 5x−3=2x+95x - 3 = 2x + 9 becomes 5x−3−2x=2x+9−2x5x - 3 - 2x = 2x + 9 - 2x.

Thinking the variable term must always go on the left

Remediation: Show examples where it's easier to collect on the right. The answer is the same either way - try both methods with the same equation.

Forgetting to distribute before combining like terms

Remediation: Use color-coding: circle the parentheses and require students to expand them as a mandatory first step before any other operations.

Differentiation Ideas

For Struggling Students:

  • • Start with equations where coefficients are small positive integers
  • • Use balance scale manipulatives or visual tools
  • • Provide step-by-step templates with fill-in-the-blank operations
  • • Focus on checking answers to build confidence

For On-Level Students:

  • • Include equations with negative coefficients
  • • Add equations requiring distribution first
  • • Introduce word problems that translate to these equations
  • • Practice recognizing identity and contradiction equations

For Advanced Students:

  • • Solve equations with fractional coefficients
  • • Work with equations containing multiple sets of parentheses
  • • Create their own word problems that lead to variables on both sides
  • • Explore systems of equations as a preview
Standards Alignment
  • 8.EE.C.7 (CCSS.MATH.CONTENT.8.EE.C.7)

    Solve linear equations in one variable

  • 8.EE.C.7a (CCSS.MATH.CONTENT.8.EE.C.7.A)

    Give examples of linear equations with one solution, infinitely many solutions, or no solutions

  • 8.EE.C.7b (CCSS.MATH.CONTENT.8.EE.C.7.B)

    Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions

Lesson Resources
  • visualBalance Scale Simulator

    See how both sides must stay balanced when solving

  • activityEquation Solver Tool

    Step-by-step guided equation solving

  • worksheetReal-World Equation Problems

    Phone plans, race problems, and more

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

When an equation has variables on both sides of the equals sign, we need to collect all variable terms on one side before solving.
The Strategy:
  1. 1.Collect variable terms on one side (usually the left)
  2. 2.Collect constant terms on the other side
  3. 3.Solve the simplified equation
Example: 5x−3=2x+95x - 3 = 2x + 9
5x−2x=9+35x - 2x = 9 + 3
3x=123x = 12
x=4x = 4
Key Principle: Whatever you do to one side, you must do to the other side to keep the equation balanced.

Worked Examples

Solve: 5x−3=2x+95x - 3 = 2x + 9
1

Identify variable terms on both sides

Left: 5x5x, Right: 2x2x → Variables on both sides
2

Subtract 2x2x from both sides

5x−2x−3=2x−2x+95x - 2x - 3 = 2x - 2x + 9 → 3x−3=93x - 3 = 9
3

Add 33 to both sides

3x−3+3=9+33x - 3 + 3 = 9 + 3 → 3x=123x = 12
4

Divide both sides by 33

3x3=123\frac{3x}{3} = \frac{12}{3} → x=4x = 4
5

Check the solution

5(4)−3=2(4)+95(4) - 3 = 2(4) + 9 gives 17=1717 = 17 \checkmark → Solution verified!

Common Mistakes

Subtracting variables incorrectly: 5x−2x=35x - 2x = 3 instead of 3x3x

Why it's wrong: When combining like terms, only the coefficients are subtracted: 5x−2x=(5−2)x=3x5x - 2x = (5-2)x = 3x.

Correct: 5x−2x=3x5x - 2x = 3x, not 33. The variable stays!

Forgetting to perform operations on BOTH sides

Why it's wrong: An equation is like a balance scale. If you only change one side, it becomes unbalanced (and wrong).

Correct: Always write out both sides: 5x−2x−3=2x−2x+95x - 2x - 3 = 2x - 2x + 9

Sign errors when moving terms: 5x=2x+125x = 2x + 12 becoming 5x−2x=125x - 2x = 12

Why it's wrong: When a term crosses the equals sign, its sign changes. But +12+12 stays positive if it's not moving.

Correct: 5x−2x=125x - 2x = 12 only if you're subtracting 2x2x from both sides.

Not distributing before collecting like terms

Why it's wrong: You cannot combine terms inside and outside parentheses until you distribute.

Correct: Always distribute first: 3(x+2)=3x+63(x + 2) = 3x + 6, then collect like terms.

Why It Matters

Equations with variables on both sides appear constantly in real-world problem solving:
  • Comparing costs: When does renting become cheaper than buying?
  • Break-even analysis: At what point do two options cost the same?
  • Distance problems: When do two objects meet if traveling toward each other?
  • Science: Balancing chemical equations, physics formulas
This skill is essential for more advanced algebra, including systems of equations and inequalities.

Real World Applications

Comparing Phone Plans

Phone companies offer different pricing structures. Equations help find the break-even point.

Example:

Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 10 euros plus 0.15 euros per minute. When do they cost the same?

1Try It Yourself

Plan A: 20 + 0.10m euros. Plan B: 10 + 0.15m euros, where m = minutes used.

At how many minutes do both plans cost the same?

Step 1: Write the mathematical expression

Set the costs equal: 20+0.10m=10+0.15m20 + 0.10m = 10 + 0.15m

Catching Up in a Race

When two runners start at different positions or speeds, we can calculate when they'll meet.

Example:

Runner A starts 50 meters ahead but runs at 6 m/s. Runner B runs at 8 m/s. When does B catch A?

2Try It Yourself

Position A: 50 + 6t meters. Position B: 8t meters (t = seconds).

After how many seconds does Runner B catch Runner A?

Step 1: Write the mathematical expression

Set positions equal: 50+6t=8t50 + 6t = 8t

Temperature Conversion

Celsius and Fahrenheit scales meet at a specific temperature.

Example:

At what temperature does Celsius equal Fahrenheit? Use F=95C+32F = \frac{9}{5}C + 32.

3Try It Yourself

If C=FC = F, then: C=95C+32C = \frac{9}{5}C + 32

At what temperature are Celsius and Fahrenheit equal?

Step 1: Write the mathematical expression

Solve: C=95C+32C = \frac{9}{5}C + 32

Key Takeaways

  • 1Equations with variables on both sides require collecting variable terms on one side first
  • 2Use inverse operations: subtract/add variable terms, then subtract/add constants
  • 3If there are parentheses, distribute before collecting like terms
  • 4Always check your solution by substituting back into the original equation
  • 5Whatever you do to one side, you must do to the other side

Frequently Asked Questions

Does it matter which side I collect the variables on?

No! You can collect variables on either side. However, collecting on the side with the larger coefficient often avoids negative coefficients, making arithmetic easier.

What if I get 0=00 = 0 as my answer?

This means the equation is an identity - it's true for ALL values of x. For example, 2x+4=2(x+2)2x + 4 = 2(x + 2) simplifies to 0=00 = 0, meaning any number works.

What if I get something like 0=50 = 5?

This means there is no solution. The equation is a contradiction. For example, x+3=x+5x + 3 = x + 5 leads to 3=53 = 5, which is never true.

Glossary

Like terms
Terms with the same variable raised to the same power (e.g., 5x5x and 2x2x are like terms)
Coefficient
The number multiplied by a variable (e.g., in 5x5x, the coefficient is 55)
Inverse operation
An operation that undoes another (addition/subtraction, multiplication/division)
Identity equation
An equation that is true for all values of the variable (e.g., x+1=x+1x + 1 = x + 1)
Contradiction
An equation with no solution because it's never true (e.g., x=x+1x = x + 1)

Formula Card

Solving Strategy

1. Distribute parentheses 2. Collect variable terms on one side 3. Collect constant terms on the other side 4. Solve and check

Steps to solve equations with variables on both sides

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