Digital SAT · Solve it in Desmos

How to Solve Systems from Word Problems on Desmos (Digital SAT)

Desmos solves the system once you have built it, handling the arithmetic and confirming the pair. Turning the words into two equations with consistent variables is the step that is yours.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for systems from word problems

The Desmos steps

1
Graph both equations
Once built, type both into Desmos in their given form.
2
Click the intersection
The crossing point gives the values satisfying both conditions.
3
Answer the asked quantity
Read the value or combination wanted from the solution.

Worked in Desmos

Easy example

At a school fair, an adult ticket costs $8 and a child ticket costs $5. A family bought

a

adult tickets and

c

child tickets, purchasing 6 tickets for a total of $39. Which system of equations represents this situation?

a+c=6

and

8a+5c=39

a+c=39

and

8a+5c=6

a+c=6

and

5a+8c=39

8a+5c=6

and

a+c=39

A: Correct. The number of tickets gives $a+c=6$ , and the total cost gives $8a+5c=39$ .

B: Incorrect. This swaps the ticket count (6) and the total cost (39) between the two equations.

C: Incorrect. This swaps the prices, attaching $5 to adult tickets and $8 to child tickets.

D: Incorrect. This both swaps the prices' roles and swaps the count and cost totals.

Explanation

The ticket count gives a + c = 6, and the total cost gives 8a + 5c = 39.

Medium example

At a parking lot, cars pay $4 and trucks pay $7. In one hour,

c

cars and

t

trucks paid, totaling 25 vehicles and $130. Which system represents this situation?

c + t = 130

and

4c + 7t = 25

c + t = 25

and

4c + 7t = 130

c + t = 25

and

7c + 4t = 130

4c + 7t = 25

and

c + t = 130

A: Incorrect. This swaps the count (25) and the revenue (130).

B: Correct. The count gives $c + t = 25$ and the revenue gives $4c + 7t = 130$ .

C: Incorrect. This attaches the wrong fee to cars and trucks.

D: Incorrect. This swaps the count and revenue totals.

Explanation

There are 25 vehicles: $c + t = 25$ . The revenue is $130: $4c + 7t = 130$ .

Hard example

A worker earns $15 per regular hour and $22 per overtime hour. In one week she worked

r

regular and

o

overtime hours, totaling 46 hours and $750. Which system represents this situation?

15r + 22o = 46

and

r + o = 750

r + o = 46

and

22r + 15o = 750

r + o = 46

and

15r + 22o = 750

r + o = 750

and

15r + 22o = 46

A: Incorrect. This swaps the hours total (46) and the pay (750).

B: Incorrect. This attaches the wrong rate to regular and overtime hours.

C: Correct. The hours give $r + o = 46$ and the pay gives $15r + 22o = 750$ .

D: Incorrect. This swaps the hours total and the pay.

Explanation

Total hours: $r + o = 46$ . Total pay: $15r + 22o = 750$ .

Detailed explanation

The hours add to 46, giving $r + o = 46$ . The pay combines the two rates: $15r + 22o = 750$ .

Try it with the calculator

Question 1easy

System of equations

The graph of a system of two linear equations is shown in the $xy$ -plane. What is the solution $(x, y)$ to this system?

Question 2easy

System of equations

Two lines are graphed in the $xy$ -plane, as shown. At what ordered pair $(x, y)$ do the lines intersect?

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.

Start the free diagnostic

Desmos questions

Can Desmos solve systems from word problems questions on the SAT?

Desmos solves the system once you have built it, handling the arithmetic and confirming the pair. Turning the words into two equations with consistent variables is the step that is yours.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master systems from word problems, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full method Take the free diagnostic

The Desmos steps

Graph both equations

Click the intersection

Answer the asked quantity

Worked in Desmos

Try it with the calculator

Question 3 is ready when you are

Desmos questions

Can Desmos solve systems from word problems questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Should I still learn the algebra?

Master systems from word problems, calculator and all