Introduction to Systems of Equations

Learn what systems of equations are and discover how two equations can work together to find a unique solution.

Intermediate25 minLesson
Printable worksheet

Definition

A system of equations is a set of two or more equations that share the same variables. The solution to a system is the set of values that makes ALL equations true at the same time.
For example, the system:
{x+y=5x−y=1\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}
has the solution x=3x = 3 and y=2y = 2, because:
  • 3+2=53 + 2 = 5 (first equation is true)
  • 3−2=13 - 2 = 1 (second equation is true)
Graphically, the solution is the point where two lines intersect.

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What is a system of equations?

Worked Examples

Is (x,y)=(4,3)(x, y) = (4, 3) a solution to the system: x+y=7x + y = 7 and 2x−y=52x - y = 5?
1

Substitute into the first equation

x+y=4+3=7x + y = 4 + 3 = 7 → First equation: 7=77 = 7 is TRUE
2

Substitute into the second equation

2x−y=2(4)−3=8−3=52x - y = 2(4) - 3 = 8 - 3 = 5 → Second equation: 5=55 = 5 is TRUE
3

Check if BOTH are satisfied

First equation: TRUE, Second equation: TRUE → Both equations are satisfied

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 (3,2)(3, 2) with (2,3)(2, 3)

Why it's wrong: In ordered pairs, the first number is xx and the second is yy. The order matters!

Correct: Always write solutions in the correct order: (x,y)(x, y). 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

Function graph

-202468-202468xy(2, 3)

Tap or hover over the graph to read off values.

y = x + 1y = -x + 5
  • Intersection point: (2, 3)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = x + 1

m=1, b=1

y = -x + 5

m=-1, b=5

Expression Calculator

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Practice Problems

16 problems
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What is a system of equations?

Why It Matters

Systems of equations help us solve problems with multiple unknowns:
  • 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
Whenever you have two unknowns and two pieces of information, systems of equations are the tool to use!

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.

1Try It Yourself

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: 2n+3p=132n + 3p = 13 and 4n+p=174n + p = 17

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.

2Try It Yourself

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 xx = liters of 40%, yy = liters of 80%. Set up: x+y=50x + y = 50 and 0.4x+0.8y=0.6(50)0.4x + 0.8y = 0.6(50)

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

If the lines are parallel (same slope but different y-intercepts), they never meet and the system has no solution. We call this an inconsistent system.
If the lines are parallel (same slope but different y-intercepts), they never meet and the system has no solution. We call this an inconsistent system.
If the two equations represent the same line, then every point on that line is a solution. This gives infinitely many solutions. We call this a dependent system.
One equation with two unknowns has infinitely many solutions (a whole line of points). The second equation narrows it down to a specific point where both conditions are met.

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
Today's challengeOne problem, three tries

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