Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Mrs. Shepherd's grandchildren are participating in a gift wrap sale to raise money for their school. She decided to stock up, so she ordered $2$ rolls of reversible paper and $5$ rolls of metallic paper from Valeria, spending a total of $\$90$ . She also ordered $4$ rolls of reversible paper and $5$ rolls of metallic paper from Jill, which cost a total of $\$120$ . Assuming that rolls of each type are priced the same, what is the price for each kind of paper? $\newline$ Rolls of reversible paper cost $\$$ _ each, and rolls of metallic paper cost $\$$ _ each.

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Mrs. Shepherd's grandchildren are participating in a gift wrap sale to raise money for their school. She decided to stock up, so she ordered

2

rolls of reversible paper and

5

rolls of metallic paper from Valeria, spending a total of

\$90

. She also ordered

4

rolls of reversible paper and

5

rolls of metallic paper from Jill, which cost a total of

\$120

. Assuming that rolls of each type are priced the same, what is the price for each kind of paper?

\newline

Rolls of reversible paper cost

\$

_____ each, and rolls of metallic paper cost

\$

_____ each.

Define variables: Define the variables for the cost of each type of paper. $\newline$ Let $x$ be the cost of one roll of reversible paper, and $y$ be the cost of one roll of metallic paper.
Write equations: Write the system of equations based on the given information. $\newline$ First order: $2$ rolls of reversible paper and $5$ rolls of metallic paper for $\$90$ . $\newline$ Second order: $4$ rolls of reversible paper and $5$ rolls of metallic paper for $\$120$ . $\newline$ The system of equations is: $\newline$ $2x + 5y = 90$ $\newline$ $4x + 5y = 120$
Eliminate variable: Decide which variable to eliminate. $\newline$ We can eliminate $y$ because it has the same coefficient in both equations.
Subtract equations: Subtract the first equation from the second to eliminate $y$ . $\newline$ $(4x + 5y) - (2x + 5y) = 120 - 90$ $\newline$ $4x + 5y - 2x - 5y = 120 - 90$ $\newline$ $2x = 30$
Solve for x: Solve for x. $\newline$ $2x = 30$ $\newline$ $x = \frac{30}{2}$ $\newline$ $x = 15$
Substitute $x$ : Substitute the value of $x$ into one of the original equations to solve for $y$ . Using the first equation: $2x + 5y = 90$ $2(15) + 5y = 90$ $30 + 5y = 90$ $5y = 90 - 30$ $5y = 60$ $y = \frac{60}{5}$ $y = 12$
Check solution: Check the solution by substituting both values into the second equation. $\newline$ $4x + 5y = 120$ $\newline$ $4(15) + 5(12) = 120$ $\newline$ $60 + 60 = 120$ $\newline$ $120 = 120$ $\newline$ The solution checks out.