Answer:
Price per mechanical pencil: [tex]\$ 0.5[/tex].
Price per ruler: [tex]\$ 1[/tex].
Step-by-step explanation:
Let the cost per mechanical pencil be [tex]\$ p[/tex], and let the cost per ruler be [tex]\$ r[/tex].
Set up two equations based on the amount of money that each teacher paid.
The cost of [tex]50[/tex] mechanical pencils ([tex]\$ p[/tex] each) and [tex]30[/tex] rulers ([tex]\$ r[/tex] each) would be [tex](50 \, p + 30\, r)[/tex]. That should be equal to the amount of money that Mrs. Ortiz paid. In other words, [tex]50 \, p + 30\, r = 55[/tex].
Similarly, the cost of [tex]30[/tex] mechanical pencils ([tex]\$ p[/tex] each) and [tex]30 \![/tex] rulers ([tex]\$ r[/tex] each) would be [tex](50 \, p + 30\, r)[/tex]. That should be equal to the amount of money that Mr. Donahue paid. In other words, [tex]30 \, p + 30\, r = 45[/tex].
That gives a system of two equations:
[tex]\left\lbrace \begin{aligned}& 50 \, p + 30\, r = 55 \\ & 30 \, p + 30\, r = 45 \end{aligned}\right.[/tex].
Subtract the second equation for the first:
[tex]20\, p = 10[/tex].
[tex]\displaystyle p = \frac{1}{2} = 0.5[/tex].
In other words, the cost of one mechanical pencil would be [tex]\$0.5[/tex].
Substitute [tex]p = 0.5[/tex] back to either equation and solve for [tex]r[/tex], the cost of one ruler.
[tex]50 \times 0.5 + 30\, r = 55[/tex].
[tex]30\, r = 55 - 25 = 30[/tex].
[tex]r = 1[/tex].
In other words, the cost of one ruler would be [tex]\$ 1[/tex].
