Question

a movie theater sells 50 tickets. adult tickets (x) cost $10 and childrens tickets (y) cost $6. the total amount collected is $400. how many childrens tickets were sold?
x + y = 50
10x + 6y = 400
a. 30
b. 20
c. 35
d. 25

if you solve a system of equations and both variables cancel out leaving a true statement like 0 = 0, what does it mean?
a. the system has infinitely many solutions.
b. the system has exactly one solution.
c. the system has no solution.
d. the equations are not linear.

after solving a system of equations, you find that x = 4 and y = 1 satisfies both 3x - 2y = 10 and x + 3y = 7. how should you check these solutions?
a. re - graph the equations.
b. solve a different system to verify.
c. solve the equations again.
d. substitute x and y back into both equations.

Explanation:

Step1: Solve the first system of equations

We have the system \(

$$\begin{cases}x + y=50\\10x + 6y=400\end{cases}$$

\). From the first equation \(x=50 - y\). Substitute \(x = 50 - y\) into the second equation:

$$
LATEXBLOCK1
$$

Step2: Recall the concept of systems of equations

For a system of linear equations \(a_1x + b_1y=c_1\) and \(a_2x + b_2y=c_2\). If after elimination (e.g., using the addition - subtraction or substitution method) we get \(0 = 0\), it means that the two equations represent the same line. So the system has infinitely many solutions.

Step3: Recall the method of checking solutions

To check if \(x = 4\) and \(y = 1\) is a solution of the system \(

$$\begin{cases}3x-2y = 10\\x + 3y=7\end{cases}$$

\), we substitute \(x = 4\) and \(y = 1\) into both equations.
For \(3x-2y\): \(3\times4-2\times1=12 - 2=10\)
For \(x + 3y\): \(4+3\times1=4 + 3=7\)

Answer:

1. d. 25
2. a. The system has infinitely many solutions.
3. d. Substitute \(x\) and \(y\) back into both equations.