Digital SAT · Quadratic & Exponential Functions

How to Solve Quadratic-Linear Systems on the Digital SAT

Q: How do you solve a system with a line and a parabola on the SAT?

Set the two expressions for y equal, solve the resulting quadratic for x, then find the matching y-values. Or graph both in Desmos and click the intersection points.

Q: How many solutions can a quadratic-linear system have?

Zero, one, or two, depending on whether the line misses the parabola, grazes it, or cuts through it. Graphing both and counting the crossings answers this at a glance.

Q: Can Desmos solve a nonlinear system?

Yes. Graph the line and the curve on separate lines and click each crossing; Desmos labels the coordinates and makes a count question trivial.

Q: How do I find where a line and a curve intersect?

The intersections are the solutions. Set the equations equal and solve, or graph both in Desmos and click the points where they meet to read the coordinates.

Q: Why do I keep missing a solution on these?

Because a line often meets a parabola in two spots and it is easy to stop after the first. A graph shows both, so the second crossing does not slip by.

Pair a line with a curve and the solutions are wherever they cross. Because a line can cut a parabola in two places, touch it once, or miss it, these questions come in a where-do-they-meet flavor and a how-many-times flavor. Graphing both in Desmos answers either instantly. By hand, set the two expressions equal, collect on one side, and solve the quadratic that results.

Written from Perfect1600’s analysis of every quadratic-linear systems question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Medium
practice questions: 73

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

0, 1, or 2

Two curves, count the crossings

The number of intersections is the number of solutions, which graphing makes obvious.

88%

Do it on Desmos

Almost all resolve by graphing both and clicking, with no algebra.

watch

Easy to miss the second

A line through a parabola can cross twice; overlooking one crossing is the classic error.

How to recognize quadratic-linear systems questions

Two equations appear, one linear and one carrying an x-squared term.
You are asked for the intersection points, a coordinate, or the number of solutions.
The phrasing mentions intersecting, solutions, or where the graphs meet.
A count version asks only how many times the two graphs cross.

Why students miss these

With two curves in play, the slips are combining them wrong or stopping short. Substitute carelessly and you solve the wrong quadratic; forget that a line can meet a parabola twice and you report one solution when there are two. The count version undoes people who try to picture it rather than graph it. Plotting both curves settles the geometry, leaving you only to read the coordinate or the count that was asked.

The step-by-step method

1
Set the two expressions equal
Both give y, so equate them to knock out y and leave one equation in x.
2
Solve the quadratic
Move every term to one side and solve; the x-values are where the graphs meet.
3
Recover the y-values
Feed each x back into the line to complete the intersection points.
4
Give what was asked
A point, a coordinate, or a count, depending on the question.

Solving it on Desmos

Graph both. Type the line and the curve as y = ... on their own lines in Desmos.
Click each crossing. Desmos labels every intersection, and the number of crossings is the number of solutions.
Read what is asked. Take the coordinate or the count straight off the graph.

Full Desmos walkthrough for quadratic-linear systems→

When to use it: For a count of solutions, graphing turns the question into simple counting, and for coordinates it beats the algebra outright. Solve by hand only when the numbers are tidy or you want to confirm an exact value.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example

The graphs of

x = 3

and

y = 2x^{2} - 5

intersect at the point

(x, y)

in the xy-plane. What is the value of

y

1

13

18

31

A: Incorrect. This computes $2(3) - 5$ , treating $x^{2}$ as $x$ .

B: Correct. Substituting $x = 3$ gives $y = 2(3)^{2} - 5 = 18 - 5 = 13$ .

C: Incorrect. This computes $2(3)^{2}$ but forgets to subtract 5.

D: Incorrect. This computes $(2 \cdot 3)^{2} - 5$ , squaring after multiplying by 2.

Explanation

Substitute $x = 3$ : $y = 2(3)^{2} - 5 = 18 - 5 = 13$ .

Medium example

How many points of intersection do the graphs of

y = 2x + 1

and

y = x^{2} + 5

have?

Exactly one

Exactly two

Infinitely many

Zero

A: Incorrect. Exactly one would require the discriminant to equal 0, but here it is negative.

B: Incorrect. Exactly two would require a positive discriminant, but here it is negative.

C: Incorrect. The graphs are a line and a parabola, which cannot overlap entirely.

D: Correct. Setting $x^{2} + 5 = 2x + 1$ gives $x^{2} - 2x + 4 = 0$ , with discriminant $4 - 16 < 0$ , so there are no real intersections.

Explanation

The equation $x^{2} - 2x + 4 = 0$ has discriminant $4 - 16 = -12 < 0$ , so there are zero intersections.

Hard example

The system

y = x^{2} - 2x

and

y = x + 4

is graphed in the xy-plane. Which ordered pair is a solution?

(4, 8)

(2, 6)

(3, 7)

(-1, 5)

A: Correct. At $(4, 8)$ : $8 = 4 + 4$ and $8 = 4^{2} - 2(4)$ , satisfying both equations.

B: Incorrect. $(2, 6)$ satisfies $y = x + 4$ but not the quadratic, since $4 - 4 = 0$ .

C: Incorrect. $(3, 7)$ satisfies $y = x + 4$ but not the quadratic, since $9 - 6 = 3$ .

D: Incorrect. $(-1, 5)$ satisfies neither, since $-1 + 4 = 3$ and $1 + 2 = 3$ .

Explanation

Setting $x^{2} - 2x = x + 4$ gives $x^{2} - 3x - 4 = 0$ , so $x = 4$ or $x = -1$ ; the point $(4, 8)$ is listed.

Detailed explanation

Setting the expressions equal gives $x^{2} - 2x = x + 4$ , or $x^{2} - 3x - 4 = 0$ . Factoring $(x - 4)(x + 1) = 0$ gives $x = 4$ or $x = -1$ , with points $(4, 8)$ and $(-1, 3)$ . Only $(4, 8)$ appears; $(-1, 5)$ uses the wrong y-value.

The common traps

Pattern	What it does	The tell
Only one of two solutions	Found one crossing and missed the second where the line cuts the parabola twice.	A line can meet a parabola twice; look for a second crossing.
Substitution slip	Combined the equations wrong and solved the wrong quadratic.	Graph both to see whether your solutions match the crossings.
Guessed the count	Estimated the number of intersections instead of graphing.	Graph both curves and count where they cross: zero, one, or two.
Gave x, not the pair	Reported an x-value when the ordered pair was wanted.	Check whether the question asks for a coordinate, a point, or a count.

Try it: two real questions

Question 1easy

The system $y = x + 2$ and $y = x^{2}$ is graphed in the xy-plane. Which ordered pair $(x, y)$ is a solution to this system?

Question 2easy

The graphs of $x = 3$ and $y = x^{2} + 1$ intersect at the point $(x, y)$ in the xy-plane. What is the value of $y$ ?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Systems of Linear Equations

Quadratic & Exponential Functions

Quadratic Equations

Nonlinear Graphs

Graph Features

Common questions

How do you solve a system with a line and a parabola on the SAT?

Set the two expressions for y equal, solve the resulting quadratic for x, then find the matching y-values. Or graph both in Desmos and click the intersection points.

How many solutions can a quadratic-linear system have?

Zero, one, or two, depending on whether the line misses the parabola, grazes it, or cuts through it. Graphing both and counting the crossings answers this at a glance.

Can Desmos solve a nonlinear system?

Yes. Graph the line and the curve on separate lines and click each crossing; Desmos labels the coordinates and makes a count question trivial.

How do I find where a line and a curve intersect?

The intersections are the solutions. Set the equations equal and solve, or graph both in Desmos and click the points where they meet to read the coordinates.

Why do I keep missing a solution on these?

Because a line often meets a parabola in two spots and it is easy to stop after the first. A graph shows both, so the second crossing does not slip by.

Practice quadratic-linear systems the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize quadratic-linear systems questions

Why students miss these

The step-by-step method

Set the two expressions equal

Solve the quadratic

Recover the y-values

Give what was asked

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you solve a system with a line and a parabola on the SAT?

How many solutions can a quadratic-linear system have?

Can Desmos solve a nonlinear system?

How do I find where a line and a curve intersect?

Why do I keep missing a solution on these?

Practice quadratic-linear systems the way it is tested