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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Two tailors, Harper and Rachel, sit down to do some embroidery. Harper can embroider shirts per hour, and Rachel can get through shirts per hour. In addition, the tailors had previously finished some shirts. Harper has already completed shirts, and Rachel has completed 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?In hours, each tailor will have finished a total of shirts.
Full solution
Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Two tailors, Harper and Rachel, sit down to do some embroidery. Harper can embroider shirts per hour, and Rachel can get through shirts per hour. In addition, the tailors had previously finished some shirts. Harper has already completed shirts, and Rachel has completed 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?In hours, each tailor will have finished a total of shirts.
- Equation for Harper: We know:Harper's embroidery rate: shirts/hourHarper's initial shirts: What is the equation for Harper's total shirts embroidered over time?Total shirts embroidered = (Embroidery rate Time) + Initial shirtsLet represent the number of hours and represent the total number of shirts embroidered.For Harper:
- Equation for Rachel: We know:Rachel's embroidery rate: shirts/hourRachel's initial shirts: What is the equation for Rachel's total shirts embroidered over time?Total shirts embroidered = (Embroidery rate Time) + Initial shirtsFor Rachel:
- Substitution to Solve: We have:Shirts embroidered by Harper: Shirts embroidered by Rachel: Use substitution to solve for .Set the two equations equal to each other since they will have embroidered the same number of shirts:
- Solving for x: Now, solve for x:Subtract from both sides:Subtract from both sides:Divide both sides by :
- Value of y: We have:y = x + What is the value of y when ?y = () + y = + y =
- Final Solution: We have:How long did it take for them to embroider the same number of shirts?And how many shirts had each of them completed?Time: hoursNumber of shirts = In hours, each tailor will have finished a total of shirts.
