At a concert, adult tickets cost \6 each. The concert sold a total of 120 tickets and collected \xy$ is the number of student tickets sold, which system of equations models this situation?

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SAT Systems of Two Linear Equations Question 5 | Free SAT Prep | Aniko

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Question 5·Medium·Systems of Two Linear Equations in Two Variables

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At a concert, adult tickets cost $12 each and student tickets cost $6 each. The concert sold a total of 120 tickets and collected $1,020 in ticket revenue. If $x$ is the number of adult tickets sold and $y$ is the number of student tickets sold, which system of equations models this situation?

For this type of SAT question, first translate each sentence directly into an equation: use $x + y = \text{(total items)}$ for total counts, and use (price per item) × (number of items) added together to equal the total cost for money statements. Then match your two equations to the answer choice that has the same coefficients (the numbers in front of $x$ and $y$ ) and the same constants (the totals), being careful not to swap prices or totals.

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Identify what x and y stand for

Ask yourself: What do $x$ and $y$ represent in this problem, and what phrase in the question talks about the total of those things?

Write an equation for the total number of tickets

Use the fact that the concert sold 120 tickets in all. How can you write an equation that shows the sum of adult and student tickets equals 120?

Write an equation for the total money collected

Each adult ticket is $12 and each student ticket is $6. How do you combine $12x$ and $6y$ so that it equals the total revenue of $1,020?

Match your equations to a choice

Once you have both equations, look for the answer option that contains exactly those two equations, with the correct numbers in the correct places.

Desmos Guide

Enter the equations from the system you chose

After you pick an answer choice, type its two equations into Desmos on separate lines (for example, equations of the form x + y = 120 and 12x + 6y = 1020). Desmos will graph each as a straight line.

Find the intersection point

Look for the point where the two lines intersect. The $x$ -coordinate of this point is the number of adult tickets, and the $y$ -coordinate is the number of student tickets that satisfy both equations at the same time.

Check that the intersection makes sense in context

Take the intersection values for $x$ and $y$ and verify two things: (1) $x + y$ equals 120 tickets, and (2) $12x + 6y$ equals $1,020. If either of these checks fails for the system you chose, that answer choice does not correctly model the situation.

Step-by-step Explanation

Translate the total number of tickets

We are told the concert sold a total of 120 tickets.

If $x$ is the number of adult tickets and $y$ is the number of student tickets, then

x + y = 120

Any correct system must include this equation, because it represents the total number of tickets sold.

Translate the total ticket revenue

Adult tickets cost $12 each and student tickets cost $6 each, and the concert collected a total of $1,020.

Money from adult tickets: $12x$
Money from student tickets: $6y$
Total money: $1,020

So the revenue equation is

12x + 6y = 1020

Any correct system must also include this equation, because it represents the total money collected.

Match both equations to the answer choices

We found the two equations that model the situation:

Total tickets: $x + y = 120$
Total revenue: $12x + 6y = 1020$

Looking at the choices, the only system that has both of these equations is

\left\{ \begin{aligned} x + y &= 120\\ 12x + 6y &= 1020 \end{aligned} \right.

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Step-by-step Explanation

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Step-by-step Explanation