Question

one - time setup fee of $10 and then $4 per box of cards. sidney, meanwhile, ordered hers online. they cost $5 per box. there was no setup fee, but she had to pay $5 to have her order shipped to her house. by coincidence, owen and sidney ended up spending the same amount on their business cards. how much did each spend? \\(\\) write a system of equations, graph them, and type the solution.

Explanation:

Step1: Define variables

Let $x$ = number of boxes, $y$ = total amount spent.

Step2: Write Owen's cost equation

Owen has a $10 setup fee + $4 per box:
$y = 4x + 10$

Step3: Write Sidney's cost equation

Sidney has a $5 shipping fee + $5 per box:
$y = 5x + 5$

Step4: Set equations equal (same cost)

$4x + 10 = 5x + 5$

Step5: Solve for $x$

Subtract $4x$ from both sides:
$10 = x + 5$
Subtract 5 from both sides:
$x = 5$

Step6: Calculate total spent

Substitute $x=5$ into $y=4x+10$:
$y = 4(5) + 10 = 20 + 10 = 30$
(Verify with $y=5x+5$: $y=5(5)+5=25+5=30$, matches)

Answer:

The system of equations is

$$\begin{cases} y = 4x + 10 \\ y = 5x + 5 \end{cases}$$

. The solution is $x=5$ (5 boxes) and each spent $\$30$.
For the graph:

• Equation $y=4x+10$: Plot the y-intercept $(0,10)$, then use slope 4 to plot $(1,14)$, $(2,18)$, etc.
• Equation $y=5x+5$: Plot the y-intercept $(0,5)$, then use slope 5 to plot $(1,10)$, $(2,15)$, etc.

The lines intersect at the point $(5, 30)$.