Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Dustin and his workout partner are lifting weights together, doing many sets of each exercise. On a certain exercise, Dustin is using a $32$ -pound bar, increasing the amount of weight he lifts by $10$ pounds on each set. His partner, meanwhile, started out using a $38$ -pound bar and is upping the weight by adding $7$ pounds on every set. Eventually, Dustin and his workout partner will be lifting the same amount, and will take turns using the same barbell. How many sets will they have completed? How much weight will they be lifting then? $\newline$ After completing $\_\_$ sets, they will both be lifting $\_\_$ pounds.

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Solve a system of equations using any method: word problems

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

\newline

Dustin and his workout partner are lifting weights together, doing many sets of each exercise. On a certain exercise, Dustin is using a

32

-pound bar, increasing the amount of weight he lifts by

10

pounds on each set. His partner, meanwhile, started out using a

38

-pound bar and is upping the weight by adding

7

pounds on every set. Eventually, Dustin and his workout partner will be lifting the same amount, and will take turns using the same barbell. How many sets will they have completed? How much weight will they be lifting then?

\newline

After completing

\_\_

sets, they will both be lifting

\_\_

pounds.

Define variables: Let's define the variables. Let $x$ represent the number of sets completed. Dustin starts with a $32$ -pound bar and adds $10$ pounds per set, so the total weight he lifts after $x$ sets is $32 + 10x$ pounds. His workout partner starts with a $38$ -pound bar and adds $7$ pounds per set, so the total weight his partner lifts after $x$ sets is $38 + 7x$ pounds. We want to find the value of $x$ when both are lifting the same amount of weight.
Write equations: Write the system of equations based on the information given. The equation for Dustin is $32 + 10x$ , and the equation for his partner is $38 + 7x$ . Since they will eventually lift the same amount, we can set these two expressions equal to each other to find the point where they intersect. $\newline$ Equation: $32 + 10x = 38 + 7x$
Solve for x: Solve the equation for x. To do this, we need to get all the x terms on one side and the constants on the other. $\newline$ $32 + 10x = 38 + 7x$ $\newline$ Subtract $7x$ from both sides: $\newline$ $32 + 10x - 7x = 38 + 7x - 7x$ $\newline$ $32 + 3x = 38$
Isolate x: Now, subtract $32$ from both sides to isolate the $x$ term. $\newline$ $32 + 3x - 32 = 38 - 32$ $\newline$ $3x = 6$
Calculate weight lifted: Divide both sides by $3$ to solve for $x$ . $\frac{3x}{3} = \frac{6}{3}$ $x = 2$
Confirm solution: Now that we have the number of sets $x$ , we can find out how much weight they will be lifting. We can plug $x$ back into either of the original equations. Let's use Dustin's equation: $\newline$ Weight lifted by Dustin after $x$ sets: $32 + 10x$ $\newline$ Weight lifted by Dustin after $2$ sets: $32 + 10(2) = 32 + 20 = 52$ pounds
Confirm solution: Now that we have the number of sets $x$ , we can find out how much weight they will be lifting. We can plug $x$ back into either of the original equations. Let's use Dustin's equation: $\newline$ Weight lifted by Dustin after $x$ sets: $32 + 10x$ $\newline$ Weight lifted by Dustin after $2$ sets: $32 + 10(2) = 32 + 20 = 52$ poundsTo confirm our solution, we should check that Dustin's partner will also be lifting $52$ pounds after $2$ sets. $\newline$ Weight lifted by Dustin's partner after $x$ sets: $38 + 7x$ $\newline$ Weight lifted by Dustin's partner after $2$ sets: $x$ $1$ pounds