A solution to a system satisfies every equation at once. Graphically, it is where the lines cross.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Graphing
Draw both lines and read the intersection. It shows what is happening, but only gives exact answers at neat coordinates.
Substitution
Solve one equation for one variable, then put that expression into the other. One equation with one unknown remains.
Choosing the variable
Pick the variable with a coefficient of 1 if there is one. It avoids fractions entirely.
Finish the job
Finding one variable is half the answer. Substitute back to get the other.
Three possibilities
Lines can cross once, be parallel with no solution, or be the same line with infinitely many.
Where two conditions meet
A solution satisfies both equations at once. Graphically it is the intersection point — the single place lying on both lines, which is why graphing makes the idea concrete.
One, none, or infinitely many
Lines cross once, are parallel, or coincide. The three outcomes correspond exactly to the three algebraic endings, and recognising which you have is part of solving.
Substitution replaces a variable
Express one variable from one equation and substitute into the other, leaving a single equation in one unknown. Choosing the equation where a variable is already isolated keeps the algebra clean.
Check in both equations
Any point on one line satisfies that equation, so checking one proves nothing. The solution must be verified in both, and this is the step most often skipped.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Elena Substitute
Elena solves y = 2x + 1 and 3x + y = 11.
- Step 1
The first equation is already solved for y, so she substitutes it into the second.
Step 4: Your Turn
Practice makes it stick.
The Substitution
Problem 1 of 2
y = 2x + 1 and 3x + y = 11. What is x?
The Other Half
Problem 2 of 2
Same system. What is y?
Find the Meeting Point
1 of 8
y = x and y = 4. What is x?
2 of 8
y = 3x and y = x + 4. What is x?
3 of 8
Same system. What is y?
4 of 8
y = 2x + 3 and y = 2x + 7. How many solutions?
5 of 8
y = 5x − 1 and 2y = 10x − 2. How many solutions?
6 of 8
x + y = 10 and x − y = 2. What is x?
7 of 8
Match each graph situation with the number of solutions.
Tap a card on the left to start.
8 of 8
y = 4x and y = 2x + 6. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 2x and y = x + 5. What is x?
Question 2 of 2
What does a solution to a system mean graphically?
What You Learned
- A solution satisfies every equation in the system at once.
- Substitution replaces one variable with an equivalent expression.
- Lines can meet once, never, or everywhere.