Solve systems at intersections

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Solve linear and nonlinear SAT systems in Desmos, interpret intersection coordinates, apply conditions, and calculate the quantity the question asks for.

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

What you’ll learn

Spot a system: two equations that share the same two unknowns.
Type each equation as it's written, without solving for $y$ first.
Read each intersection that matters as a solution $(x,y)$ .
Use a condition to pick the right solution.
Work out a coordinate or an expression from that solution.
Turn two facts from a word problem into a system.
Tell when substitution or elimination is quicker by hand.

Prerequisites

Why this matters on the SAT

Turn two equations into one answer

SAT systems can come with awkward coefficients, equations in different forms, or a line paired with a curve. Desmos handles all of those the same way. Graph both equations, and any point where the graphs meet makes both equations true at once.

SAT example

The solution to the system

\begin{aligned} 7x-4y&=16\\[1.4em] 3x+5y&=27 \end{aligned}

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

Fast Desmos solution

Type both equations the way they're written. The lines cross at

(4,3).

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

and the answer is C.

Look at that last step. The graph gave you $x$ and $y$ , but the question wanted $x+y$ . The SAT might ask for the whole pair, one coordinate, or something built from both. So finding the point is only half the job. The other half is simple to say and easy to forget: find the point, then read the question.

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The lines cross at (4, 3), the one point on both lines.

When should you solve a system in Desmos?

Desmos is at its best when two equations share the same two unknowns and solving them by hand would take a while.

Start with Desmos

Reach for Desmos when solving by hand would take several steps, like the awkward coefficients in $7x-4y=16$ and $3x+5y=27$ .
Graph it when one equation is a line and the other is a curve, like $y=2x+3$ and $y=(x-2)^2+4$ .
If the graphs might meet more than once, a graph shows you every meeting point together.
Desmos also suits questions that want $x$ , $y$ , the pair, or something built from both, like $x+y$ , since one point gives you both coordinates.

Use algebra instead

Solve by hand when substitution or elimination takes one quick step. For $x+y=10$ and $x-y=4$ , adding the equations gives $2x=14$ , so $(x,y)=(7,3)$ with almost no work.

You’ll spot these questions by their wording: “the solution is $(x,y)$ ,” “satisfies both equations,” or “for the solution with $x>0$ .”

Check your understanding:

Here are two systems: $x+y=12$ and $x-y=2$ , or $y=x^2-4$ and $y=3x+6$ . Which one would you put into Desmos, and which would you solve by hand? Why?

Enter both equations as written

You don't have to solve for $y$ first. Desmos can graph an equation like $2x+3y=13$ as it is, so type each equation on its own line, in its original form.

Take the system

\begin{aligned} 2x+3y&=13\\[1.4em] 5x-y&=7. \end{aligned}

Type 2x+3y=13 on one line and 5x-y=7 on the next. The lines cross at

(2,3).

Why is that point the solution? Put in $x=2$ and $y=3$ , and both equations come out true:

2(2)+3(3)=13

and

5(2)-3=7.

Try it yourself:

Change the $7$ in the second equation to $11$ . Before you look, guess which way the crossing point will move. Then click the new point to check your guess.

Check your understanding:

Why does an intersection have to make both equations true, not only one of them?

Common mistake:

Rearranging when you didn’t need to, and slipping on a sign. Solving $5x-y=7$ for $y$ should give $y=5x-7$ , but it’s easy to write $y=7-5x$ instead. If Desmos can graph the original form, type that and skip the risk.

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Both equations, typed as written. Click where the lines cross.

Read what the question asks for

When you solved one-variable equations, $y$ was only a helper for comparing the two sides. In a system, $x$ and $y$ are both unknowns you're solving for. So one point, like

(2,3),

can answer several different questions:

If the question asks for	You submit
the solution $(x,y)$	$(2,3)$
the value of $x$	$2$
the value of $y$	$3$
the value of $x+y$	$5$
the value of $3x-2y$	$0$

So after you click the intersection, read the question one more time. The point is what you work with. It isn't always what you submit.

Common mistake:

Flipping the coordinates. In $(2,3)$ , $x$ always comes first, so $x=2$ and $y=3$ .

Check your understanding:

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

One more thing about the numbers you see. Desmos labels points with numbers, not exact forms like fractions or roots. If a coordinate shows up as a decimal that never ends, but the question wants an exact value, don't guess which fraction or root it stands for. Switch it to a fraction if that gives a clear result, check an answer choice, or work that part out with exact algebra. Exactness, rounding, and output verification goes deeper on exact answers.

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

Spot the condition and what to submit

There are two solutions here, and the question wants only one. The condition $x>3$ tells you which. Once you've found that point, you'll submit its $y$ -coordinate.

Step 2

Graph both equations

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

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

Before you choose, make sure you can see every place the graphs meet. If a curve runs off the edge of the screen, zoom out, because another intersection could be hiding outside the window.

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Find both meeting points before you choose one.

Step 3

Use the condition

Check each point against $x>3$ . The point $(1,5)$ has $x=1$ , which is too small. The point

(5,13)

has $x=5$ , so it's the one you want.

Step 4

Submit the coordinate the question asks for

At $(5,13)$ , $y=13$ , so the answer is

\boxed{13}.

Common mistake: Submitting $5$ . You just used the $x$ -coordinate to pick the point, so it's the number on your mind. But the question asks for $y$ .

Some questions don't ask which solution. They ask how many there are, or which constant gives none, one, or infinitely many. That's a counting job, and it comes later, in How many solutions?

Turn two facts into a system

Many SAT word problems give you two facts about the same two quantities. Give each quantity a letter, then turn each fact into its own equation.

Say a rectangle has a perimeter of $50$ units, and its length is $7$ units more than its width. Call the length $\ell$ and the width $w$ :

The perimeter gives $2\ell+2w=50$ .
The length being $7$ more than the width gives $\ell-w=7$ .

Desmos graphs with $x$ and $y$ , so type $x$ wherever you have $\ell$ and $y$ wherever you have $w$ , and keep that match the whole way through. The lines cross at $(16,9)$ , so the rectangle is $16$ units long and $9$ units wide.

Before you submit, check the units and ask whether the numbers make sense for the situation. Both of these are positive, as side lengths have to be. A count of things, like animals or boxes, has to be a whole number that isn't negative.

Check your understanding:

A farm has $28$ chickens and goats in total, and together they have $80$ legs. Let $c$ be the number of chickens and $g$ the number of goats. What system describes the farm?

Common mistake:

Putting the same total in both equations, like writing $2c+4g=28$ for the farm. The $28$ counts animals, not legs. Each equation needs its own fact from the problem: animals in one, legs in the other.

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Here x is the length and y is the width.

Finish the solution

Both lines are already graphed. Click where they cross, then work out what the question asks for.

Finish the solution

The solution to the system

\begin{aligned} 3x+y&=13\\[1.4em] x-2y&=9 \end{aligned}

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

First steps

Both equations are typed in and graphed.
Click the intersection and read both coordinates.

Finish it

Your answer

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Read the point, then work out x + y.

Practice problems

Your turn. Each problem takes a graph and one careful last step. Try each one before you open its hint.

Practice problem

The solution to the system

\begin{aligned} y&=3x-5\\[1.4em] 2x+y&=10 \end{aligned}

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

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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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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

Type each equation on its own line, as it's written. Desmos can graph many systems without solving for $y$ first.
Every intersection is a solution: a point that makes both equations true.
The first coordinate is $x$ and the second is $y$ .
When the graphs might meet more than once, look at enough of the graph to see every meeting point.
Use every condition, like $x>3$ or a count that can't be negative, before you pick a point.
Find the point, then read the question. It may want $x$ , $y$ , the pair, or an expression like $x+y$ .
For a word problem, give each quantity a letter and turn each fact into its own equation.
When substitution or elimination is clearly shorter, solve by hand.

Next lesson

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

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Practice

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3,300 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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