SAT Systems of Two Linear Equations Question 64 | Free SAT Prep | Aniko

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Question 64·Easy·Systems of Two Linear Equations in Two Variables

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At a school fundraiser, adult tickets cost $5 each and student tickets cost $3 each. Suppose $a$ represents the number of adult tickets sold and $s$ represents the number of student tickets sold. If 40 tickets were sold in total and $160 was collected, which system of equations represents this situation?

For "which system of equations" word problems, first clearly define the variables, then write one equation that counts the total quantity of items (like tickets, people, or objects) and a second equation that tracks a value related to them (like money, distance, or time) using the given per-item rates. Pay close attention to units: plain variables usually count items, while coefficients (like 5 or 3) usually come from prices or rates; make sure counts equal totals such as 40 tickets, and money expressions equal dollar totals such as 160 dollars, then choose the answer that matches both equations.

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Start with the total number of tickets

Focus on the sentence "40 tickets were sold in total." Using $a$ for adult tickets and $s$ for student tickets, how can you write an equation that shows their sum is 40?

Now model the money collected

Each adult ticket brings in $5 and each student ticket brings in $3. How do you express the total money from $a$ adult tickets and $s$ student tickets, and what should that total equal?

Match units to equations

One equation should involve just counting tickets (no 5 or 3 as prices) and equal the total tickets, and the other should use 5 and 3 as prices and equal the total dollars. Look for the choice where 40 is a ticket total and 160 is a money total.

Desmos Guide

Set up variables in Desmos

Treat the horizontal axis variable $x$ as the number of adult tickets and the vertical axis variable $y$ as the number of student tickets when you enter equations into Desmos.

Graph one answer choice at a time

Pick an answer choice. Rewrite its equations by replacing $a$ with $x$ and $s$ with $y$ , then type those two equations on separate lines in Desmos to graph the two lines for that choice.

Check whether the intersection satisfies the word conditions

Click on the intersection point of the two lines to see its coordinates $(x,y)$ . Using those coordinates, compute $x + y$ (this should match the total number of tickets, 40) and $5x + 3y$ (this should match the total money, 160). If either value is not 40 or 160 respectively, that answer choice does not correctly represent the situation.

Step-by-step Explanation

Use the total number of tickets

We are told that 40 tickets were sold in total and that $a$ is the number of adult tickets and $s$ is the number of student tickets.

So, the number of adult tickets plus the number of student tickets must equal 40 tickets. This gives the "ticket-count" equation in words:

(number of adult tickets) + (number of student tickets) = 40.

Use the total amount of money

Adult tickets cost $5 each and student tickets cost $3 each.

Money from adult tickets is 5 dollars times the number of adult tickets, which is $5a$ .
Money from student tickets is 3 dollars times the number of student tickets, which is $3s$ .

The total money collected is $160, so in words we have:

(money from adult tickets) + (money from student tickets) = 160 dollars.

That means the sum of $5a$ and $3s$ must equal 160.

Write the equations and match the answer choice

Now turn the word equations into algebraic equations:

Ticket-count equation: adult tickets + student tickets = 40 becomes
$a + s = 40$ .
Money equation: $5a$ (adult money) + $3s$ (student money) = 160 becomes
$5a + 3s = 160$ .

So the system that represents the situation is

\begin{aligned} a + s &= 40 \\ 5a + 3s &= 160 \end{aligned}

This matches answer choice A.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation