Introduction to Systems of Equations
Learn what systems of equations are and discover how two equations can work together to find a unique solution.
Definition
- (first equation is true)
- (second equation is true)
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Worked Examples
Is a solution to the system: and ?
Substitute into the first equation
→ First equation: is TRUE
Substitute into the second equation
→ Second equation: is TRUE
Check if BOTH are satisfied
First equation: TRUE, Second equation: TRUE → Both equations are satisfied
Answer: Yes, is a solution because it satisfies both equations.
Common Mistakes
Checking only one equation and declaring it a solution
Why it's wrong: A solution must satisfy ALL equations in the system, not just one.
Correct: Always substitute your answer into every equation to verify it works for all of them.
Confusing the solution with
Why it's wrong: In ordered pairs, the first number is and the second is . The order matters!
Correct: Always write solutions in the correct order: . Label your work clearly.
Thinking parallel lines have a solution
Why it's wrong: Parallel lines never intersect, so there's no point that lies on both lines.
Correct: If two lines are parallel (same slope, different intercept), the system has no solution.
Interactive Visual
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Balance Scale
Solution: x = 4
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y = 2x + 1
m=2, b=1
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Practice Problems
16 problemsWhat is a system of equations?
Why It Matters
- Business: Determining the price of items when given total costs from different orders
- Chemistry: Balancing chemical equations and mixture problems
- Engineering: Calculating forces in structures with multiple support points
- Everyday life: Figuring out how many of each item someone bought when you know the total count and total cost
Real World Applications
Shopping Problems
Determining individual prices when you only know combined totals.
Example:
If 3 apples and 2 oranges cost 11 euros, and 2 apples and 3 oranges cost 9 euros, you can find the price of each fruit using a system of equations.
A store sells notebooks and pens. Emma buys 2 notebooks and 3 pens for 13 euros. Tom buys 4 notebooks and 1 pen for 17 euros.
What is the price of one notebook?
Step 1: Write the mathematical expression
Set up the system: and
Mixture Problems
Combining different substances to achieve a target concentration or quantity.
Example:
Mixing a 10% acid solution with a 30% acid solution to create 100 mL of a 22% solution.
A chemist needs 50 liters of a 60% alcohol solution. She has a 40% solution and an 80% solution available.
How many liters of the 80% solution does she need?
Step 1: Write the mathematical expression
Let = liters of 40%, = liters of 80%. Set up: and
Key Takeaways
- 1A system of equations is two or more equations with the same variables
- 2The solution is the values that make ALL equations true simultaneously
- 3Graphically, the solution is the intersection point of the lines
- 4To verify a solution, substitute the values into each equation and check
- 5Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line)
Frequently Asked Questions
Glossary
- System of equations
- A set of two or more equations that share the same variables and are solved together
- Solution of a system
- The ordered pair(s) that satisfy all equations in the system simultaneously
- Consistent system
- A system that has at least one solution
- Inconsistent system
- A system with no solution (parallel lines)
- Dependent system
- A system where the equations represent the same line, giving infinitely many solutions