Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two tailors, Harper and Rachel, sit down to do some embroidery. Harper can embroider $4$ shirts per hour, and Rachel can get through $6$ shirts per hour. In addition, the tailors had previously finished some shirts. Harper has already completed $21$ shirts, and Rachel has completed $5$ shirts. Harper and Rachel decide to take a break when they have finished the same total number of shirts. How long will that take? How many shirts, in total, will each tailor have finished? $\newline$ In $\underline{\quad}$ hours, each tailor will have finished a total of $\underline{\quad}$ shirts.

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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 tailors, Harper and Rachel, sit down to do some embroidery. Harper can embroider

4

shirts per hour, and Rachel can get through

6

shirts per hour. In addition, the tailors had previously finished some shirts. Harper has already completed

21

shirts, and Rachel has completed

5

shirts. Harper and Rachel decide to take a break when they have finished the same total number of shirts. How long will that take? How many shirts, in total, will each tailor have finished?

\newline

\underline{\quad}

hours, each tailor will have finished a total of

\underline{\quad}

shirts.

Equation for Harper: We know: $\newline$ Harper's embroidery rate: $4$ shirts/hour $\newline$ Harper's initial shirts: $21$ $\newline$ What is the equation for Harper's total shirts embroidered over time? $\newline$ Total shirts embroidered = (Embroidery rate $\times$ Time) + Initial shirts $\newline$ Let $x$ represent the number of hours and $y$ represent the total number of shirts embroidered. $\newline$ For Harper: $y = (4 \times x) + 21$
Equation for Rachel: We know: $\newline$ Rachel's embroidery rate: $6$ shirts/hour $\newline$ Rachel's initial shirts: $5$ $\newline$ What is the equation for Rachel's total shirts embroidered over time? $\newline$ Total shirts embroidered = (Embroidery rate $\times$ Time) + Initial shirts $\newline$ For Rachel: $y = (6 \times x) + 5$
Substitution to Solve: We have: $\newline$ Shirts embroidered by Harper: $y = 4x + 21$ $\newline$ Shirts embroidered by Rachel: $y = 6x + 5$ $\newline$ Use substitution to solve for $x$ . $\newline$ Set the two equations equal to each other since they will have embroidered the same number of shirts: $\newline$ $4x + 21 = 6x + 5$
Solving for x: Now, solve for x: $\newline$ $4x + 21 = 6x + 5$ $\newline$ Subtract $4x$ from both sides: $\newline$ $21 = 2x + 5$ $\newline$ Subtract $5$ from both sides: $\newline$ $16 = 2x$ $\newline$ Divide both sides by $2$ : $\newline$ $x = 8$
Value of y: We have: $\newline$ y = $4$ x + $21$ $\newline$ What is the value of y when $x = 8$ ? $\newline$ y = $4$ ( $8$ ) + $21$ $\newline$ y = $32$ + $21$ $\newline$ y = $53$
Final Solution: We have: $\newline$ $x = 8$ $\newline$ $y = 53$ $\newline$ How long did it take for them to embroider the same number of shirts? $\newline$ And how many shirts had each of them completed? $\newline$ Time: $8$ hours $\newline$ Number of shirts = $53$ $\newline$ In $8$ hours, each tailor will have finished a total of $53$ shirts.