Hello. In the previous lesson, you solved one-step and two-step equations by preserving equality and applying inverse operations. You also checked solutions by substitution. This lesson extends that same balance principle to multi-step linear equations: first simplify each side, then gather variable terms on one side and constants on the other.
By the end, you should be able to solve equations that include parentheses, like terms, and variables on both sides of the equals sign—and recognize when an equation has no solution or is true for every value.
The central strategy
A linear equation may look more complicated than the equations you have solved so far, but the destination is still the same:
For many multi-step equations, the work has a reliable order:
- Simplify each side separately. Distribute into parentheses and combine like terms.
- Choose a variable side. Get all terms containing the variable onto one side.
- Move constants to the other side.
- Divide or multiply so the coefficient of the variable is 1.
- Check the answer in the original equation.
Every line must represent an equivalent equation. When you add, subtract, multiply, or divide, do it to both sides. When you distribute or combine like terms, you simplify the side where those terms already appear.
A useful distinction:
- , because both are like terms on the same side.
- You cannot combine on the left with on the right. Instead, use an equality-preserving operation to collect them on one side.
Variables on both sides
Consider:
There are variable terms on both sides. We can choose either side to hold all the variable terms. Here, subtracting from both sides keeps the remaining variable coefficient positive:
Now remove the from the variable side by adding 7 to both sides:
Finally, divide both sides by 2:
The solution is
Notice that “moving to the other side” is shorthand. What actually happened was this:
That is why the on the right becomes zero and the on the left becomes .
In the Step-by-step solution image, the equation becomes
then
and finally
It is completely acceptable for the variable term to end up on the right temporarily. An equation such as
contains the same information as
A complete example with parentheses and like terms
Before collecting variables, simplify any expressions that are not already in their simplest form. Solve:
The parentheses on the left must be removed first. Distribute 5 to both terms inside:
Now combine the like terms on the left:
At this point, the equation has variables on both sides. Subtract from both sides:
Add 20 to both sides:
Divide by 4:
So the solution is
This is the kind of equation where the sequence matters: distribute first, combine like terms second, collect variable terms third, and only then finish with inverse operations.
Solving Multi-Step Equations | Distributive Property, Combining Like Terms, Variables on Both Sides
Watch “Solving Multi-Step Equations | Distributive Property, Combining Like Terms, Variables on Both Sides” by Math with Mr. J. It works through this same complete structure and makes the decision-making order visible.
Watch the full example. Begin with the roadmap, which identifies the questions to ask before doing any algebra. Then follow simplifying first: distribution and combining like terms happen before terms are collected across the equation. Watch isolating the variable for the balanced operations that produce the solution, and finish with the check. While watching, write down the equation after each major stage: distribute, combine, collect variables, collect constants, and divide.
The check confirms the result using the original equation:
Substitute :
Evaluate the left side:
Evaluate the right side:
Both sides equal 76, so is correct.
Distribution requires careful signs
Parentheses often create the extra steps in a multi-step equation. The distributive property says:
The multiplier outside the parentheses affects every term inside. In particular, a negative multiplier changes signs.
For example,
becomes
because
and
Now solve:
First distribute:
Then combine constants on the left:
Subtract from both sides:
Add 7 to both sides:
Check the original equation:
Therefore,
The main risk in this type of problem is not the final division; it is a sign mistake during distribution or while combining negative terms. Writing every intermediate line is usually faster than repairing an error later.
When the usual answer does not exist
Most equations in this lesson have one solution. But sometimes the variable terms cancel completely. What remains tells you whether the original equation is always true or never true.
Infinitely many solutions
Consider:
Distribute and combine:
Subtract from both sides:
This statement is true. It does not restrict at all: whatever number replaces , both sides are equal. Therefore, the equation has infinitely many solutions.
You can state the answer as:
No solution
Now consider:
Distribute and combine:
Subtract from both sides:
This is false. No value of can make a false numerical statement become true. Therefore, the equation has no solution.
Do not try to divide after the variable terms have canceled. The correct conclusion comes from the final numerical statement:
| Final result after simplifying | Meaning |
|---|---|
| One solution | |
| Infinitely many solutions; all real numbers work | |
| No solution |
A dependable checklist
When an equation appears complicated, do not search for a shortcut first. Use this checklist:
- Look for parentheses. Distribute carefully, including negative signs.
- Combine like terms on each side. Variables combine with variables; constants combine with constants.
- Choose where the variable should remain. If possible, choose an operation that leaves a positive coefficient.
- Use addition or subtraction on both sides to collect all variable terms together.
- Use addition or subtraction on both sides to place constants on the opposite side.
- Divide by the variable’s coefficient.
- Check in the original equation.
Decimals and fractions do not change this logic. For example, in
you would subtract , subtract 2, and then divide by . The arithmetic may be less comfortable, but the structure is identical.
Common habits to avoid:
| Avoid | Use instead |
|---|---|
| Combining terms across the equals sign | Combine like terms only on the same side. |
| Saying a term “crosses over and changes sign” without a reason | State the operation: add or subtract the same term on both sides. |
| Dividing before removing a separate constant term | First isolate the variable term, then divide by its coefficient. |
| Distributing to only one term in parentheses | Multiply the outside factor by every term inside. |
| Stopping when variables disappear | Decide whether the resulting numerical statement is true or false. |
| Checking a simplified equation only | Substitute into the original equation. |
Key takeaways
Multi-step linear equations use the same equality principle as one-step and two-step equations. The extra work is organizational:
- Simplify each side first by distributing and combining like terms.
- Collect variable terms on one side and constants on the other.
- Isolate the variable using inverse operations.
- Check the solution in the original equation.
- If variables cancel, a true statement means infinitely many solutions; a false statement means no solution.
You have now completed the Algebra Foundations module. The next module shifts from solving equations to representing relationships visually, beginning with ordered pairs on the Cartesian coordinate plane.
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