Substitution Method
Learn to solve systems of equations by substituting one equation into another.
Definition
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Worked Examples
Solve the system:
Identify the isolated variable
The first equation already has isolated: →
Substitute into the other equation
Replace with in the second equation: →
Solve for
→
Back-substitute to find
→
Verify the solution
Check in second equation: ✓ → Solution verified!
Answer: The solution is
Common Mistakes
Forgetting to distribute when substituting
Why it's wrong: When you substitute an expression like and there's a coefficient in front, you must distribute.
Correct: If you have , distribute to get , not .
Substituting into the same equation
Why it's wrong: If you solve equation 1 for and substitute back into equation 1, you'll just get (always true) and learn nothing.
Correct: Always substitute into the OTHER equation to create a new equation with one variable.
Sign errors when distributing negatives
Why it's wrong: Subtracting an expression like means multiplying each term by .
Correct: , not .
Not checking the solution in BOTH equations
Why it's wrong: A solution must satisfy both equations. Checking only one equation doesn't guarantee correctness.
Correct: Always substitute your answer into both original equations to verify.
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Practice Problems
16 problemsIn the system , what expression should you substitute for in the second equation?
Why It Matters
- Exact answers: Unlike graphing, substitution gives precise solutions, not estimates
- Works for all systems: Can solve systems that are hard to graph accurately
- Foundation for advanced math: The same substitution technique is used in calculus, physics, and engineering
- Real-world applications: Comparing phone plans, mixing solutions, and balancing budgets all use this method
Real World Applications
Comparing Phone Plans
Phone companies offer different plans. The substitution method helps find when two plans cost the same.
Example:
Plan A costs 20 euros monthly plus 0.10 euros per minute. Plan B costs 30 euros monthly plus 0.05 euros per minute. When are they equal?
Plan A: (cost for minutes) Plan B:
After how many minutes do both plans cost the same?
Step 1: Write the mathematical expression
Set the costs equal and solve:
Mixing Solutions in Chemistry
Scientists often need to mix solutions of different concentrations to get a desired result.
Example:
A chemist needs 100 mL of a 30% acid solution. They have 20% and 50% solutions available.
Let = mL of 20% solution, = mL of 50% solution. Total volume: Acid content:
How much of each solution is needed?
Step 1: Write the mathematical expression
From the first equation: . Substitute:
Budget Planning
Businesses use systems of equations to balance costs and revenue.
Example:
A company sells basic and premium products. If they sell 50 items and make 2000 euros, with basic at 30 euros and premium at 50 euros, how many of each did they sell?
Let = basic products, = premium products. Total items: Total revenue:
How many of each product type was sold?
Step 1: Write the mathematical expression
From the first equation: . Substitute:
Key Takeaways
- 1The substitution method solves systems by replacing one variable with an equivalent expression
- 2Choose the variable with coefficient 1 or that's already isolated for easiest solving
- 3Always substitute into the OTHER equation, not the one you solved
- 4After finding one variable, back-substitute to find the other
- 5Verify your solution by checking both original equations
Frequently Asked Questions
Glossary
- Substitution method
- A technique for solving systems where you replace a variable with an equivalent expression from another equation
- System of equations
- Two or more equations with the same variables that must be solved simultaneously
- Back-substitution
- Plugging a found value back into an equation to find the remaining variable
- Solution to a system
- The ordered pair that makes both equations true
Formula Card
Substitution Method Steps
Follow these five steps to solve any system using substitution