Solve systems at intersections

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Difficulty: Beginner
Estimated time: 35 minutes
Domains: Algebra Advanced Math
Techniques: SystemsIntersections

What you’ll learn

Recognize a system with two equations and two original unknowns.
Enter both equations directly without rearranging either one first.
Read every relevant intersection as an ordered-pair solution.
Apply a condition to select the required solution.
Calculate a coordinate or expression from the solution.
Translate two contextual conditions into a system.
Choose a shorter algebra method when substitution or elimination is immediate.

Prerequisites

Why this matters on the SAT

Turn two equations into one answer

SAT systems can contain awkward coefficients, equations in different forms, or a line paired with a curve. Desmos graphs both conditions together. Every intersection is a point that makes both equations true.

SAT example

The solution to the system

\begin{aligned} 7x-4y&=16\\ 3x+5y&=27 \end{aligned}

is $(x,y)$ . What is the value of $x+y$ ?

A
$5$
B
$6$
C
$7$
D
$8$

Fast Desmos solution

Enter both equations exactly as written. You do not need to solve either one for $y$ . The graphs intersect at $(4,3)$ , so

x+y=4+3=\boxed{7}.

The answer is C.

Finding the intersection is only the calculator step. The SAT may ask for the full ordered pair, one coordinate, or an expression built from both coordinates.

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The intersection is the ordered pair that satisfies both equations.

When should you solve a system in Desmos?

Desmos is strongest when two conditions share the same two unknowns and graphing avoids slow substitution or elimination.

Begin with Desmos when…

two equations involve the same two unknowns;
the equations would take several algebra steps;
one equation is linear and the other is nonlinear;
the graphs may intersect more than once; or
the prompt asks for $x$ , $y$ , an ordered pair, or an expression such as $x+y$ .

Use algebra when…

substitution or elimination is immediate. For $x+y=10$ and $x-y=4$ , adding gives $2x=14$ , so $(x,y)=(7,3)$ with very little work.

Look for wording such as “the solution is $(x,y)$ ,” “satisfies both equations,” or “for the solution with $x>0$ .”

Check your understanding:

Which system is a stronger reason to begin with Desmos: $x+y=12$ and $x-y=2$ , or $y=x^2-4$ and $y=3x+6$ ? Explain the difference.

1. Enter both equations as written

Desmos can graph equations that are not solved for $y$ . Keep each original equation on its own expression line.

For

\begin{aligned} 2x+3y&=13\\ 5x-y&=7, \end{aligned}

enter 2x+3y=13 and 5x-y=7. The lines intersect at $(2,3)$ .

That point is the solution because substituting $x=2$ and $y=3$ makes both equations true:

2(2)+3(3)=13

and

5(2)-3=7.

Try it yourself:

Change the $7$ in the second equation to $11$ . Predict how the intersection will move, then select the new point to check.

Check your understanding:

Why must an intersection satisfy both equations rather than only one?

Common mistake:

Rearranging an equation unnecessarily and introducing a sign error. If Desmos can graph the original form, enter it directly.

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Enter each equation in its original form and select the intersection.

2. Read the requested quantity

In Lesson 3, $y$ was sometimes a temporary comparison value. In a two-variable system, both $x$ and $y$ are original unknowns. The intersection $(2,3)$ can answer several different questions:

Question	Answer
What is the solution $(x,y)$ ?	$(2,3)$
What is the value of $x$ ?	$2$
What is the value of $y$ ?	$3$
What is the value of $x+y$ ?	$5$
What is the value of $3x-2y$ ?	$0$

Read the prompt again after selecting the intersection. Use the ordered pair as input for the final calculation rather than assuming the entire point is the answer.

Common mistake:

Swapping the coordinates. In $(2,3)$ , the first coordinate is $x=2$ and the second is $y=3$ .

Check your understanding:

The solution to a system is $(2,3)$ . What is the value of $4x+y$ ?

Graph labels are numerical. If an intersection is displayed as a nonterminating decimal but the question requires an exact value, do not guess its symbolic form. Use fraction conversion when it gives a clear result, verify an answer choice, or use exact algebra. Lesson 27 develops the full exactness and output-verification workflow.

3. Example: Apply a condition to multiple intersections

Worked example

The graphs of

y=2x+3

and

y=(x-2)^2+4

intersect at two points.

For the solution $(x,y)$ with $x>3$ ,

what is the value of $y$ ?

Step 1

Identify the condition and target

The system has two intersections. Keep both visible, then use $x>3$ to select one. The final answer is that point’s $y$ -coordinate.

Step 2

Graph both equations

Enter y=2x+3 and y=(x-2)^2+4. The graphs intersect at

(1,5)\qquad\text{and}\qquad(5,13).

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Keep both intersections visible before applying the condition.

Step 3

Apply the condition

The point $(1,5)$ does not satisfy $x>3$ . The point $(5,13)$ does, so it is the required solution.

Step 4

Submit the requested coordinate

At the selected point, $y=13$ . Therefore,

\boxed{13}.

Common mistake:

Submitting $5$ , the selected point’s $x$ -coordinate, when the question asks for $y$ .

This lesson uses visible intersections to solve a system and apply a stated condition. When the main question is how many solutions exist or which parameter creates none, one, or infinitely many solutions, use the full solution-count workflow.

4. Translate two conditions into a system

Many SAT word problems describe two conditions involving the same two quantities. Define one variable for each quantity, then write one equation for each condition.

Suppose a rectangle has a perimeter of $50$ units, and its length is $7$ units greater than its width. If $\ell$ is the length and $w$ is the width, then:

the perimeter condition becomes $2\ell+2w=50$ ;
the side-length condition becomes $\ell-w=7$ .

Desmos uses $x$ and $y$ as graphing variables, so replace $\ell$ with $x$ and $w$ with $y$ consistently. The equations intersect at $(16,9)$ , so the rectangle is $16$ units long and $9$ units wide. Both values are positive, as side lengths must be.

After solving a contextual system, check the units and restrictions. Counts must be nonnegative whole numbers, and measurements must be valid for the situation.

Check your understanding:

A farm has $28$ chickens and goats in total. Together they have $80$ legs. If $c$ is the number of chickens and $g$ is the number of goats, what system represents the situation?

Common mistake:

Using the same total in both equations. Each equation must represent a different condition from the problem.

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The x-coordinate represents the length, and the y-coordinate represents the width.

Finish the solution

The system is already graphed. Select the intersection, then submit the expression the question asks for.

Read, then add

Finish the solution

The solution to the system

\begin{aligned} 3x+y&=13\\ x-2y&=9 \end{aligned}

is $(x,y)$ . What is the value of $x+y$ ?

First steps

Both equations are entered and their graphs are visible.
Select the intersection and read both coordinates.

Finish it

Your answer

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Read the ordered pair, then evaluate the requested expression.

Practice problems

SAT practice problems

Enter each equation on its own line. Inspect every relevant intersection, apply any condition, and submit only the quantity the question asks for.

Read the requested coordinate

Practice problem

The solution to the system

\begin{aligned} y&=3x-5\\ 2x+y&=10 \end{aligned}

is $(x,y)$ . What is the value of $y$ ?

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Compare two intersections

Practice problem

The graphs of

y=x+4

and

y=x^2-2x+4

intersect at points $(x,y)$ in the $xy$ -plane. What is the greater possible value of $y$ ?

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Build a contextual system

Practice problem

A distributor packed $150$ cartons for a shipment. Each large carton held $12$ bottles, and each small carton held $8$ bottles. The cartons held $1{,}560$ bottles in total.

How many small cartons were packed?

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Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Enter each equation on its own expression line. Desmos can graph many systems without first solving for $y$ .
Every intersection is an ordered pair that satisfies both equations.
The first coordinate is $x$ , and the second is $y$ .
Inspect the relevant graph broadly when more than one intersection may exist.
Apply every sign or context condition before selecting a solution.
Calculate the coordinate or expression the question requests.
Translate two contextual conditions into two equations.
Use algebra when substitution or elimination is clearly shorter.

Next lesson

Evaluate, combine, and compose functions

Define functions once, evaluate them efficiently, and calculate combinations and compositions.

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Free Practice

What you’ll learn

Prerequisites

Turn two equations into one answer

Fast Desmos solution

When should you solve a system in Desmos?

Begin with Desmos when…

Use algebra when…

1. Enter both equations as written

2. Read the requested quantity

3. Example: Apply a condition to multiple intersections

Identify the condition and target

Graph both equations

Apply the condition

Submit the requested coordinate

4. Translate two conditions into a system

Finish the solution

Read, then add

First steps

Finish it

Practice problems

SAT practice problems

Read the requested coordinate

Compare two intersections

Build a contextual system

4 practice examples left

Quick recap

Evaluate, combine, and compose functions

Free question bank practice problems