Systems of Equations: The Graphical View
Graphically, the solution to a system is the single point where the two lines cross — the coordinates that make both equations true.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
The intersection point is the solution. Parallel lines (same slope, different intercept) never cross, so the system has no solution; identical lines overlap everywhere, giving infinitely many.
How to solve one, step by step
Example: where do $y = 2x + 1$ and $y = -x + 4$ meet?
- Set them equal: $2x + 1 = -x + 4$.
- Solve: $3x = 3 \Rightarrow x = 1$, then $y = 3$.
- They intersect at $(1, 3)$.
The mistakes that cost points
- Confusing the intersection with an intercept. The solution is where the lines meet each other, not where either crosses an axis.
- Missing parallel lines. Equal slopes mean no intersection — no solution, not one.
- Reading only one coordinate. The solution is the full point $(x, y)$.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A line has slope $4$ and passes through the point $(2, 10)$. Which of the following is the equation of the line in slope-intercept form?
- A$y = 4x - 18$
- B$y = 4x - 2$
- C$y = 4x + 18$
- D$y = 4x + 2$
Show solution
Answer: D, $y = 4x + 2$. Use point-slope form: $y - y_1 = m(x - x_1)$. Substituting $m = 4$, $x_1 = 2$, $y_1 = 10$: $y - 10 = 4(x - 2)$. Distribute: $y - 10 = 4x - 8$. Add 10 to both sides: $y = 4x + 2$.
Medium
A straight line passes through the point $(2, 4)$ and has a slope of $-5$. Which of the following statements about the line is correct?
- AThe equation is $y = -5x + 14$ and the line also passes through $(5, -11)$
- BThe equation is $y = -5x + 4$ and the line also passes through $(1, -1)$
- CThe equation is $y = -5x - 6$ and the line also passes through $(-2, 4)$
- DThe equation is $y = 2x - 5$ and the line also passes through $(2, -1)$
Show solution
Answer: A, The equation is $y = -5x + 14$ and the line also passes through $(5, -11)$. Use slope-intercept form $y = mx + b$ with the given slope $-5$. Substitute the point $(2, 4)$: $4 = -5(2) + b$, so $4 = -10 + b$, so $b = 4 + 10 = 14$. The equation is $y = -5x + 14$.
Hard
A straight line passes through the point $(4, 13)$ and has a slope of $\frac{7}{4}$. Which of the following statements about the line is correct?
- AThe equation is $y = \frac{7}{4}x + 6$ and the line also passes through $(12, 27)$
- BThe equation is $y = \frac{7}{4}x + 13$ and the line also passes through $(12, 34)$
- CThe equation is $y = \frac{7}{4}x + 20$ and the line also passes through $(-4, 13)$
- DThe equation is $y = \frac{13}{4}x + \frac{7}{4}$ and the line also passes through $(4, \frac{59}{4})$
Show solution
Answer: A, The equation is $y = \frac{7}{4}x + 6$ and the line also passes through $(12, 27)$. Use slope-intercept form $y = mx + b$ with the given slope $\frac{7}{4}$. Substitute the point $(4, 13)$: $13 = \frac{7}{4}(4) + b$, so $13 = 7 + b$, so $b = 13 - 7 = 6$. The equation is $y = \frac{7}{4}x + 6$.
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