Solving Linear Equations
2. Equations with Variables on Both Sides
Learning outcomes
- I can identify equations that contain variables on both sides.
- I can simplify equations by combining like terms.
- I can move variable terms to one side of an equation.
- I can solve equations with variables on both sides accurately.
- I can check and verify solutions.
Introduction
In earlier lessons, you learned to solve equations where the variable appeared on only one side of the equal sign. However, many real-world problems produce equations in which the variable appears on both sides. These equations require one additional step: moving all the variable terms to one side before solving.
Although these equations may look more complicated, they follow exactly the same principle as simpler equations—you must keep the equation balanced by performing the same operation on both sides. Once all the variables are collected on one side, the equation becomes familiar and can be solved using inverse operations.
What Are Equations with Variables on Both Sides?
An equation with variables on both sides has one or more variable terms on each side of the equal sign.
Examples:
Unlike one-step or two-step equations, you must first collect all the variable terms together before solving.
Step 1 – Simplify Each Side
If either side contains like terms, combine them before solving.
Like terms have exactly the same variable raised to the same power.
For example:
Example
Simplify:
Combine the like terms:
This makes the equation easier to solve.
Visualising Like Terms
Like-term tiles help show why only terms with the same variable can be combined.
Step 2 – Move the Variable Terms
Choose one side to contain all the variables.
Most students find it easiest to move the smaller variable term.
Worked Example 1
Solve:
Step 1
Subtract x from both sides.
Step 2
Subtract 4.
Step 3
Divide by 2.
Step 3 – Solve the Remaining Equation
Once the variables are on one side, solve using the same methods learned previously.
Worked Example 2
Solve:
Subtract
2x:
Subtract 7:
Divide by 3:
Example with Variables on Both Sides and Negative Numbers
Negative numbers require careful attention to signs.
Worked Example 3
Solve:
Subtract 2x:
Add 8:
Divide by 4:
Always perform the same operation on both sides to keep the equation balanced.
Example Requiring Like Terms First
Sometimes you must simplify before moving variables.
Worked Example 4
Solve:
Step 1
Combine like terms.
Step 2
Subtract x.
Step 3
Subtract 5.
Step 4
Divide by 4.
Checking Your Solution
Always substitute your answer back into the original equation.
Example
Original equation:
Solution:
Check the left side:
Check the right side:
Both sides equal 13, so the solution is correct.
Special Cases
Not every equation has exactly one solution.
Identity
Example:
Subtract 4x:
This statement is always true.
Answer: Every value of x is a solution.
No Solution
Example:
Subtract 3x:
This statement is never true.
Answer: No solution exists.
Summary Table
| Final Result | Meaning |
|---|---|
| One solution | |
| Infinitely many solutions (identity) | |
| No solution |
Common Mistakes
Students often make these errors.
❌ Moving a term without changing its sign.
✔ Remember that adding or subtracting a term changes its position only because you perform the operation on both sides.
❌ Forgetting to combine like terms first.
✔ Simplify each side whenever possible before solving.
❌ Making arithmetic mistakes with negative numbers.
✔ Work carefully and write each step clearly.
❌ Forgetting to check the final answer.
✔ Substitute the value into the original equation, not a simplified version.
Real-World Connection
Equations with variables on both sides appear in many practical situations. For example, a mobile phone company may charge a monthly fee plus a cost per gigabyte of data. Comparing two different pricing plans involves setting the total costs equal and solving for the amount of data used. Engineers, scientists, economists, and computer programmers frequently solve equations of this type when comparing competing models or determining when two quantities become equal.
Did You Know?
Many computer algebra systems, including those used by engineers and scientists, solve equations by following the same logical steps you learn in school: simplify each side, collect like terms, isolate the variable, and verify the solution. Although computers perform these steps much faster, they are applying the same mathematical principles.
Key Terms
- Equation — a mathematical statement showing that two expressions are equal.
- Variable — a symbol representing an unknown value.
- Like terms — terms with the same variable raised to the same power.
- Simplify — combine like terms or reduce an expression to its simplest form.
- Inverse operations — operations that undo each other.
- Identity — an equation that is true for every value of the variable.
- No solution — an equation with no value that satisfies it.
- Solution — a value that makes an equation true.
Key Takeaways
- Equations with variables on both sides require collecting all variable terms on one side before solving.
- Always simplify each side by combining like terms first.
- Use inverse operations to move variables and constants while keeping the equation balanced.
- Some equations have one solution, some have infinitely many solutions, and some have no solution.
- Always verify your answer by substituting it into the original equation.