# A $2 raffle ticket offers a bonus $1 early bird draw. 400 tickets were sold for the draw and a total of $894 was collected from ticket sales. How many tickets were bought for $2 and how many were...

A $2 raffle ticket offers a bonus $1 early bird draw. 400 tickets were sold for the draw and a total of $894 was collected from ticket sales. How many tickets were bought for $2 and how many were bought for $3?

System of equations

I know I have to make a system of equations and solve the variables but I am having trouble figuring out what the equations ( and variables) would be?

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### 1 Answer

Define the variables.

x = tickets bought for $2

y = tickets bought for $3

400 tickets were sold, so x + y = 400.

$894 was collected. x is worth $2 each and y is worth $3 each, so 2x + 3y = 894

Your system is:**x + y = 400**

**2x + 3y = 894**

Solve the first equation for x.

x + y = 400

x = 400 - y

Now substitute this expression in for x in the second equation.

2x + 3y = 894

2(400 - y) + 3y = 894

Now solve for y.

2(400 - y) + 3y = 894

800 - 2y + 3y = 894

800 + y = 894

y = 94

Use this value to find x in the first equation.

x + y = 400

x + 94 = 400

x = 306

**Solution: {x = 306, y = 94}**

You can check this answer by substituting both values into the second equation.

2x + 3y = 894

2 * 306 + 3 * 94 = 894

612 + 282 = 894

894 = 894

The solution is correct.

**Solution: {x = 306, y = 94}**

**306 tickets were bought for $2 and 94 tickets were bought for $3.**