Question

1. for admission to a concert at the state fair, child tickets cost x dollars and adult tickets cost y dollars. solve the system shown to find the price of each type of ticket. a solve the system by graphing to find the price of each ticket. \\(\

$$\begin{cases} x + y = 10 \\\\ 4x + 8y = 64 \\end{cases}$$

\\) b what does the solution mean?

Explanation:

Step1: Rewrite equations in slope-intercept form

For $x + y = 10$:
$y = -x + 10$
For $4x + 8y = 64$, simplify first:
Divide by 4: $x + 2y = 16$, then $y = -\frac{1}{2}x + 8$

Step2: Find intercepts for first line

For $y = -x + 10$:
x-intercept: set $y=0$, $0 = -x + 10 \implies x=10$ (point $(10,0)$)
y-intercept: set $x=0$, $y=10$ (point $(0,10)$)

Step3: Find intercepts for second line

For $y = -\frac{1}{2}x + 8$:
x-intercept: set $y=0$, $0 = -\frac{1}{2}x + 8 \implies x=16$ (point $(16,0)$)
y-intercept: set $x=0$, $y=8$ (point $(0,8)$)

Step4: Solve algebraically to verify

Substitute $y = 10 - x$ into $4x + 8y = 64$:
$4x + 8(10 - x) = 64$
$4x + 80 - 8x = 64$
$-4x = -16 \implies x=4$
Then $y = 10 - 4 = 6$

(Part B):
The solution $(4,6)$ represents the price of each type of ticket: the $x$-value is the cost of a child ticket ($\$4$), and the $y$-value is the cost of an adult ticket ($\$6$). It is the point where the two lines of the system intersect, meaning this price pair satisfies both equations from the problem's context.

Answer:

(Part A):
A child ticket costs $\$4$, an adult ticket costs $\$6$.