Question

Determine which system below will produce infinitely many solutions.

-4x + 6y = 20
4x - 3y = -2

-2x + 4y = 18
4x - 8y = -36

3x - 2y = 5
-6x - 4y = -10

5x - 3y = 12
15x - 9y = 24

The art museum had a total of 224 visitors on Tuesday. Visitors older than 18 paid $12 for admission. Visitors 18 years or younger paid $8 for admission. The museum collected a total of $2,520. How many visitors were 18 or younger?

42

56

182

194

Answer 1

-2x + 4y = 18
4x - 8y = -36

This system has infinite solutions as one equation is a factor of the other

for the second we can check that
a number of visitors older than 18
b number of visitors younger than 18

a(12) +b(8) = 2520
a + b = 224

solving the equation system we have
that 182 visitors were 18 or younger

Answer 2

First task:
To determine this we need to make difference between system that doesn't have solution and system that has infinite number of solution. If when solving equations we get 0 = 0 that means system has infinite number of equations.
If we get 0 = some number that means that system doesn't have solution.

Multiplying first equation of second system by 2 and adding it to second equation we get 0=0 which means that this system has infinite number of solutions.

Answer is second system

Second task:
For this we need to set a system of equations and solve it. System looks like this:
x + y = 224
x*12 + y*8=2520
-------------------------
x represents older than 18 and "y" 18 and younger than that.
first equation we multiply with -12 and sum it with second.
-4y = -168
y=168/4 = 42

Answer is 42.