SAT Systems of Two Linear Equations Question 2 | Free SAT Prep | Aniko

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Question 2·Medium·Systems of Two Linear Equations in Two Variables

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At a school fundraiser, adult tickets cost $7 each and student tickets cost $3 each. If 200 tickets were sold for a total of $1,040, how many adult tickets were sold?

For ticket and money word problems, quickly define variables for each type of ticket, then write one equation for the total number of tickets and another for the total amount of money. Put the system in a simple form (like $a+s=\text{total tickets}$ ) that makes elimination easy—often you can multiply the simpler equation and subtract to eliminate one variable in one step. On questions like this, you can also reason numerically: imagine all tickets at the cheaper price, compare that total to the actual total, and divide the extra money by the price difference between adult and student tickets to get the number of adult tickets. This combination of algebra and numerical insight saves time and reduces arithmetic mistakes on the SAT.

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Identify the unknowns

Decide what letters will represent the number of adult tickets and the number of student tickets. How can you express the fact that 200 tickets were sold in total using these letters?

Write the money equation

Use the prices: adult tickets cost 7 dollars and student tickets cost 3 dollars. How can you write an equation for the total of 1,040 dollars using your variables?

Solve the system efficiently

You should now have two equations. Try to eliminate one variable: notice that one equation already has $a+s$ . How could you multiply it so the $s$ terms in both equations match and can be subtracted?

Desmos Guide

Enter the equations for the system

Let $x$ represent the number of adult tickets and $y$ the number of student tickets. In Desmos, type the two equations in slope–intercept form:

y = 200 - x (from $x+y=200$ )
y = (1040 - 7x)/3 (from $7x+3y=1040$ ).

Find the intersection point

Zoom or adjust the visible window until you can clearly see where the two lines cross. Click or tap on the intersection point. The $x$ -coordinate of this point is the number of adult tickets sold, and the $y$ -coordinate is the number of student tickets sold.

Step-by-step Explanation

Define variables and write the equations

Let $a$ be the number of adult tickets and $s$ be the number of student tickets.

Total tickets: $a+s=200$ .
Total money: each adult ticket is 7 dollars and each student ticket is 3 dollars, so

7a+3s=1040.

You now have a system of two equations in two variables.

Use elimination to remove one variable

Start with the system:

\begin{aligned} a+s&=200 \\ 7a+3s&=1040 \end{aligned}

Multiply the first equation by 3 so the $s$ -terms match:

\begin{aligned} 3(a+s)&=3(200) \\ 3a+3s&=600. \end{aligned}

Now subtract this new equation from the money equation:

\begin{aligned} (7a+3s)-(3a+3s)&=1040-600 \\ 4a&=440. \end{aligned}

This gives a single equation with only $a$ .

Solve for the number of adult tickets

From the previous step you have

4a=440.

Divide both sides by 4:

a=\frac{440}{4}.

This fraction is the number of adult tickets sold.

Evaluate and check the solution

Compute the value:

a=\frac{440}{4}=110.

So 110 adult tickets were sold. As a quick check, that means $s=200-110=90$ student tickets. The total money is

110\cdot 7+90\cdot 3=770+270=1040,

which matches the fundraiser total, confirming that 110 is correct.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation