Systems of Linear Equations • Topic 1 of 4

Graphical Method

Graphing each equation as a line, the solution of the system is the point where the lines cross, because that point lies on both lines. If the lines are parallel they never meet, so there is no solution; if they coincide, every point works and there are infinitely many solutions. Comparing slopes and intercepts tells you which case you are in without plotting: equal slopes and different intercepts mean parallel. On the Digital SAT, the built-in Desmos calculator makes the graphical method fast — type both equations and read the intersection.

✅ Solved examples

1. The lines y = x + 1 and y = −x + 5 cross where?
Set x + 1 = −x + 5: 2x = 4, x = 2, then y = 3. Solution (2, 3).
2. How many solutions does a system of two parallel lines have?
Parallel lines never intersect, so there is no solution.
3. y = 2x and y = 2x + 3: how many solutions?
Same slope, different intercept → parallel → no solution.
4. Where do y = 3x and y = x + 4 meet?
3x = x + 4 → 2x = 4 → x = 2, y = 6. Solution (2, 6).

✏️ Practice — try these, take hints as needed

1. Where do y = x + 2 and y = −x + 6 cross?
Set the right sides equal.
x + 2 = −x + 6 → 2x = 4.
Find x then y.
(2, 4).
2. How many solutions for two identical lines?
They lie on top of each other.
Every point works.
—
Infinitely many.
3. y = 4x + 1 and y = 4x − 2: how many solutions?
Compare slopes and intercepts.
Same slope, different intercept.
Parallel.
None.
4. Where do y = 2x − 1 and y = x + 2 meet?
Set equal: 2x − 1 = x + 2.
x = 3.
Find y.
(3, 5).
5. y = 5x and y = 5x: how many solutions?
The equations are identical.
Same line.
—
Infinitely many.

📝 Topic test — 8 questions

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