Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two classmates, Billy and Evelyn, plan to meet in the computer lab to type up their research papers. Billy can type at a speed of $1$ page per hour, whereas Evelyn can type $3$ pages per hour. So far, Billy already has $14$ pages typed up, compared to Evelyn's $4$ pages. Once they sit down and start typing together, the two students will reach the same page count before too long. How long will that take? What will the page count be? $\newline$ After $\_$ hours, each student will have a page count of $\_$ pages.

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Solve a system of equations using substitution: word problems

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

\newline

Two classmates, Billy and Evelyn, plan to meet in the computer lab to type up their research papers. Billy can type at a speed of

1

page per hour, whereas Evelyn can type

3

pages per hour. So far, Billy already has

14

pages typed up, compared to Evelyn's

4

pages. Once they sit down and start typing together, the two students will reach the same page count before too long. How long will that take? What will the page count be?

\newline

After

\_

hours, each student will have a page count of

\_

pages.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of hours they type in the computer lab. $\newline$ Let $y$ be the total number of pages each student has after $x$ hours. $\newline$ Billy's typing rate is $1$ page per hour, and he starts with $14$ pages. So, the equation for Billy's total pages after $x$ hours is: $\newline$ $y = 1 \times x + 14$ $\newline$ $y = x + 14$
Billy's Equation: Evelyn's typing rate is $3$ pages per hour, and she starts with $4$ pages. So, the equation for Evelyn's total pages after $x$ hours is: $\newline$ $y = 3 \times x + 4$
Evelyn's Equation: Now we have two equations: $\newline$ $1$ . $y = x + 14$ (Billy's equation) $\newline$ $2$ . $y = 3x + 4$ (Evelyn's equation) $\newline$ Since we are looking for the point where they have the same number of pages, we can set the two equations equal to each other and solve for $x$ : $\newline$ $x + 14 = 3x + 4$
Set Equations Equal: Subtract $x$ from both sides to get the $x$ terms on one side: $\newline$ $x + 14 - x = 3x + 4 - x$ $\newline$ $14 = 2x + 4$
Solve for x: Subtract $4$ from both sides to isolate the term with $x$ : $\newline$ $14 - 4 = 2x + 4 - 4$ $\newline$ $10 = 2x$
Substitute $x$ : Divide both sides by $2$ to solve for $x$ : $\frac{10}{2} = \frac{2x}{2}$ $5 = x$
Final Result: Now that we have the value of $x$ , we can substitute it back into either equation to find $y$ . Let's use Billy's equation: $\newline$ $y = x + 14$ $\newline$ $y = 5 + 14$ $\newline$ $y = 19$
Final Result: Now that we have the value of $x$ , we can substitute it back into either equation to find $y$ . Let's use Billy's equation: $\newline$ $y = x + 14$ $\newline$ $y = 5 + 14$ $\newline$ $y = 19$ We have found that after $5$ hours, each student will have a page count of $19$ pages.