How many solutions?

Your feedback matters:

Lesson progressPractice problems 0/4

Difficulty: Intermediate
Estimated time: 45 minutes
Domains: Algebra Advanced Math
Techniques: TangencySolution-count

What you’ll learn

Connect the number of graph intersections to the number of real solutions.
Distinguish zero, one, two, and infinitely many solutions.
Recognize line-parabola tangency as exactly one intersection.
Distinguish distinct parallel lines from identical lines.
Test exact parameter values to create a stated number of solutions.
Verify delicate graph results with exact relationships.
Choose algebra when it proves the solution count faster.

Prerequisites

Why this matters on the SAT

Count solutions without solving every coordinate

SAT questions may ask how many solutions an equation or system has without asking for the solutions themselves. In Desmos, every intersection represents one real solution. Count the intersections, not every intercept or notable point on either graph.

SAT example

The graphs of

y=x^2-4x+3

and

y=x-1

are shown in the $xy$ -plane. How many solutions does the corresponding system have?

A
Zero
B
Exactly one
C
Exactly two
D
Infinitely many

Fast Desmos solution

Enter both equations on separate lines. The graphs intersect at $(1,0)$ and $(4,3)$ . Two intersections mean two ordered-pair solutions, so the answer is C.

The coordinates confirm where the solutions are, but the question asks only for their number. Stop after you have counted and verified both intersections.

Calculator loads as you approach

Each intersection is one solution to the system.

When should you count intersections in Desmos?

Use the graph when its intersections make the solution count easier to classify. Use exact algebra when a short relationship proves the count immediately.

Begin with Desmos when…

the problem directly asks for the number of real solutions;
a nonlinear equation is easier to compare as two graphs;
a line and a curve may cross, touch, or miss each other;
a nonlinear problem gives exact choices for one unknown constant; or
the graph makes a transition between solution counts visible.

Use algebra when…

two lines can be classified by slopes or proportional coefficients;
a vertex gives the count immediately;
a quadratic simplifies to a square such as $(x-a)^2=0$ ; or
an exact fractional parameter would be awkward to locate visually.

Look for wording such as “How many real solutions?”, “exactly one solution,” “no solution,” “infinitely many solutions,” “intersect at exactly one point,” or “for which value of $k$ does the system have...?” Do not use regression to count solutions. Regression fits parameter values; it does not preserve and display every solution to an equation or system.

Check your understanding:

A system contains two linear equations whose coefficients can be compared in one step. A different system pairs a parabola with a line and gives four values of $k$ to test for exactly one solution. Which method is stronger for each problem?

1. Match intersections to solutions

The graph pattern translates directly:

How the graphs meet	Number of real solutions
No intersections	$0$
One intersection	$1$
Two intersections	$2$
More intersections	The number of intersections
Complete overlap	Infinitely many

For a single equation, graph the left side and right side as separate expressions. The $x$ -coordinates where those graphs meet are the real solutions, so count the distinct intersections.

For the line-and-parabola cases in this lesson, a line that meets the parabola at exactly one point is tangent to it.

Consider

y=(x-1)^2+2.

The horizontal line $y=1$ gives zero intersections, $y=2$ touches the vertex once, and $y=5$ crosses the parabola twice.

Try it yourself:

Hide two horizontal lines and predict the count for the remaining pair before looking at the graph. Then switch lines and repeat.

One visible intersection does not always prove there is only one. Another intersection may lie outside the current window. Pan or zoom to inspect the relevant graph broadly, especially when a curve leaves the screen.

Check your understanding:

Two graphs have no intersection in the current window. Can you conclude that the system has zero solutions? What should you do next?

Common mistake:

Counting an $x$ -intercept, $y$ -intercept, vertex, or other point that belongs to only one graph. Count only points shared by both conditions.

Calculator loads as you approach

Hide and show the horizontal lines one at a time to compare zero, one, and two solutions.

2. Separate parallel from identical

Two different lines can intersect at most once:

different slopes give one solution;
the same slope with different intercepts gives zero solutions; and
the same line written in two forms gives infinitely many solutions.

The calculator contains

y=2x+1,

the parallel line

y=2x-3,

and the identical equation

2y=4x+2.

The first and second lines never meet. The first and third equations overlap completely.

Try it yourself:

Keep $y=2x+1$ visible and toggle $2y=4x+2$ . The visible line should change color without moving. Then add y=-x+4 and notice that a different slope creates one intersection.

For equations in standard form, compare the complete equations:

if the variable coefficients are proportional but the constants are not, the lines are distinct and parallel;
if every coefficient and constant has the same multiplier, the equations represent the same line; and
if the variable coefficients are not proportional, the lines intersect once.

Check your understanding:

The first equation is $4x-3y=5$ . How many solutions result when the second equation is $12x-9y=15$ ? How does the answer change if the second constant is $18$ instead?

Common mistake:

Seeing only one line and concluding that the system has one solution. One visible line may represent two identical equations with infinitely many shared points.

This lesson uses overlap only to classify infinitely many solutions. General answer-choice testing for equivalent expressions belongs to Equivalent expressions by graph overlap.

Calculator loads as you approach

Toggle the first and third expressions to confirm that their graphs occupy the same line.

3. Example: Create exactly one intersection

Use the slider workflow without reteaching it: enter the equations, type each candidate value exactly, count intersections, and verify the winner.

Worked example

The system of equations

y=x^2-4x+k

and

y=2x+1

has exactly one solution.

Which choice gives the value of $k$ ?

A
$8$
B
$9$
C
$10$
D
$11$

Step 1

Translate the condition

Exactly one solution means the line and parabola must share one point. In this case, the line is tangent to the parabola.

Step 2

Graph both equations

Enter both equations and add one slider for $k$ . Keep both graphs visible while testing the choices.

Calculator loads as you approach

Type each candidate value for k and count the intersections.

Step 3

Test exact candidates

At $k=8$ and $k=9$ , the graphs intersect twice. At $k=11$ , they do not intersect. At $k=10$ , the graphs touch once at $(3,7)$ , so choice C creates exactly one solution.

Step 4

Verify the tangency

Set the equations equal at $k=10$ :

x^2-4x+10=2x+1.

Then

x^2-6x+9=0

and

(x-3)^2=0.

There is one distinct real value, $x=3$ , so the system has exactly one solution. The answer is C.

Common mistake:

Accepting a value because the graphs look almost tangent. Type the candidate exactly and use a vertex, substitution, or a quadratic that simplifies to $(x-a)^2=0$ to confirm a delicate visual match.

For two simple lines, coefficient comparison is often faster than testing a slider. For a line and a curve, graphing exact candidates makes the zero-to-one-to-two transition easier to see.

Finish the solution

The equations and slider are already entered. Predict the constant from the coefficient relationship, use the graph to verify complete overlap, and submit the constant.

Create infinitely many solutions

Finish the solution

The system

\begin{aligned} 3x-2y&=7\\ 6x-4y&=c \end{aligned}

has infinitely many solutions, where $c$ is a constant. What is the value of $c$ ?

First steps

Compare the variable coefficients in the two equations.
Determine the multiplier from the first equation to the second.
Apply the same multiplier to the constant $7$ .
Type the resulting value of $c$ and confirm that the two graphs overlap completely.

Finish it

Your answer

Calculator loads as you approach

Use the coefficient relationship to predict c, then verify complete overlap.

Practice problems

SAT practice problems

Count only distinct real solutions. Use graphing for nonlinear shape and candidate testing. Use short exact algebra when proportional coefficients or another exact relationship makes the count immediate.

Classify two lines

Practice problem

How many solutions does the system have?

\begin{aligned} 4x-6y&=11\\ 10x-15y&=30 \end{aligned}

Calculator loads as you approach

Create exactly one intersection

Practice problem

The graphs of

y=(x+2)^2+3

and

y=k

intersect at exactly one point. What is the value of $k$ ?

Your answer

Calculator loads as you approach

Graph both sides and count

Practice problem

How many distinct real solutions does the equation

(x+2)^4=(x+2)^2

have?

Calculator loads as you approach

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Count intersections between the two conditions: none means zero solutions, one means one solution, and complete overlap means infinitely many.
A line tangent to a parabola has exactly one intersection; a line that crosses it may have two.
Distinct parallel lines have zero solutions, while identical equations have infinitely many.
Inspect the relevant graph broadly, type parameter candidates exactly, and verify delicate results with an exact relationship.
Use algebra when a coefficient comparison, a vertex, or a quadratic that simplifies to $(x-a)^2=0$ settles the count faster.

Next lesson

Tables, lists, and calculated columns

Organize values, create calculated columns, and inspect patterns efficiently in Desmos.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Count solutions without solving every coordinate

Fast Desmos solution

When should you count intersections in Desmos?

Begin with Desmos when…

Use algebra when…

1. Match intersections to solutions

2. Separate parallel from identical

3. Example: Create exactly one intersection

Translate the condition

Graph both equations

Test exact candidates

Verify the tangency

Finish the solution

Create infinitely many solutions

First steps

Finish it

Practice problems

SAT practice problems

Classify two lines

Create exactly one intersection

Graph both sides and count

4 practice examples left

Quick recap

Tables, lists, and calculated columns

Free question bank practice problems