Systems of Equations (Substitution)

Learn to solve systems of two equations using the substitution method.

Advanced25 minLesson

Definition

A system of equations is a set of two or more equations with the same variables. The substitution method solves the system by:

1. Isolating one variable in one equation 2. Substituting that expression into the other equation 3. Solving for the remaining variable 4. Back-substituting to find the other variable

For example, to solve:

Substitute into the second equation:

Then find :

Solution:

Try it now

In the system , which equation should you substitute INTO?

Worked Examples

Solve the system:

Identify the isolated variable

The first equation already has isolated: → Ready to substitute

Substitute into the second equation

Replace with in : →

Solve for

→

Back-substitute to find

→

Verify the solution

Check in both equations: ✓ ✓ → Solution verified

Answer:

Common Mistakes

Forgetting to distribute when substituting

Why it's wrong: When substituting an expression like , the parentheses indicate the entire expression replaces the variable.

Correct: Always use parentheses and distribute: , not

Substituting back into the wrong equation

Why it's wrong: If you substitute back into the equation you used for substitution, you might get a trivial identity instead of a value.

Correct: After finding one variable, substitute into the OTHER original equation or the isolation equation.

Sign errors when isolating variables

Why it's wrong: Moving terms across the equals sign requires changing signs.

Correct: From , we get (not )

Not verifying the solution in both equations

Why it's wrong: The solution must satisfy both equations simultaneously. Checking only one equation may miss errors.

Correct: Always substitute into BOTH original equations to verify.

Interactive Visual

Linear Function Explorer

y = x

Slope (m)1

Y-Intercept (b)0

run

rise

x	y
-2	-2
-1	-1
0	0
1	1
2	2

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Equation type:Multi-step

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y = 2x + 1

m=2, b=1

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Practice Problems

15 problems

Problem 1 of 15

Easy

In the system , which equation should you substitute INTO?

Why It Matters

Systems of equations model real situations where multiple constraints exist simultaneously:

Business: Finding break-even points where cost equals revenue
Chemistry: Balancing mixtures with different concentrations
Physics: Determining where two objects meet
Economics: Finding market equilibrium between supply and demand

The substitution method is especially efficient when one equation already has a variable isolated, making it the fastest approach in many practical problems.

Real World Applications

Business Break-Even Analysis

Companies use systems of equations to find where revenue equals costs.

Example:

If cost (500 dollars fixed plus 10 dollars per item) and revenue (25 dollars per item), break-even is when : , so items.

1Try It Yourself

A bakery has fixed costs of 200 dollars per day. Each cake costs 8 dollars to make and sells for 20 dollars.

How many cakes must be sold to break even?

Step 1: Write the mathematical expression

Set cost equal to revenue:

Mixture Problems

Scientists and pharmacists use systems to create solutions with specific concentrations.

Example:

To make 100 mL of 30% acid solution from 20% and 50% solutions: Let = mL of 20%, = mL of 50%. Then and .

2Try It Yourself

A chemist needs 200 mL of 40% alcohol solution. She has 30% and 60% solutions available.

How much of each solution should she mix?

Step 1: Write the mathematical expression

Write the system: and

Key Takeaways

1The substitution method solves systems by replacing one variable with an equivalent expression
2Choose to isolate the variable with the simplest coefficient (ideally 1 or -1)
3Use parentheses when substituting expressions and distribute carefully
4Back-substitute to find the second variable after solving for the first
5Always verify your solution in BOTH original equations

Frequently Asked Questions

Use substitution when one equation already has a variable isolated (like ) or when a variable has coefficient 1 or -1. Use elimination when variables have coefficients that are easy to match or cancel.

Glossary

System of equations: Two or more equations with the same variables that must be solved simultaneously
Substitution: Replacing a variable with an equivalent expression from another equation
Solution to a system: The ordered pair that satisfies all equations in the system
Isolate: Rewrite an equation to get one variable alone on one side
Back-substitute: Substitute a found value back into an equation to find the other variable

Formula Card

Step 1: Isolate

(or solve for )

Solve one equation for one variable

Step 2: Substitute

Replace that variable in the other equation

Use parentheses around the expression

Step 3: Solve

Solve the resulting equation

Now there's only one variable

Step 4: Back-substitute

Plug value into isolation equation

Find the other variable

Step 5: Verify

Check in both equations

Both equations must be true