Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Hunter and his workout partner are lifting weights together, doing many sets of each exercise. On a certain exercise, Hunter is using a $35$ -pound bar, increasing the amount of weight he lifts by $2$ pounds on each set. His partner, meanwhile, started out using a $41$ -pound bar and is upping the weight by adding $1$ pound on every set. Eventually, Hunter 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 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

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

35

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

2

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

41

-pound bar and is upping the weight by adding

1

pound on every set. Eventually, Hunter 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: Define variables for the number of sets completed and the weight lifted. $\newline$ Let $x$ represent the number of sets completed by both Hunter and his partner. $\newline$ Let $y$ represent the weight lifted by both after $x$ sets.
Write Hunter's Equation: Write the equation for Hunter based on the given information. $\newline$ Hunter starts with a $35$ -pound bar and adds $2$ pounds per set. $\newline$ The equation for Hunter is: $y = 2x + 35$ .
Write Partner's Equation: Write the equation for Hunter's partner based on the given information. $\newline$ Hunter's partner starts with a $41$ -pound bar and adds $1$ pound per set. $\newline$ The equation for Hunter's partner is: $y = x + 41$ .
Set Up System of Equations: Set up the system of equations to find when Hunter and his partner will be lifting the same amount of weight. $\newline$ The system of equations is: $\newline$ $y = 2x + 35$ $\newline$ $y = x + 41$
Use Substitution: Use substitution to solve the system of equations. $\newline$ Since both equations equal $y$ , set them equal to each other to find $x$ : $\newline$ $2x + 35 = x + 41$
Solve for x: Solve for x by isolating the variable. $\newline$ Subtract $x$ from both sides of the equation: $\newline$ $2x + 35 - x = x + 41 - x$ $\newline$ $x + 35 = 41$
Substitute $x$ into Equation: Subtract $35$ from both sides to find the value of $x$ . $\newline$ $x + 35 - 35 = 41 - 35$ $\newline$ $x = 6$
Verify Solution: Substitute the value of $x$ back into one of the original equations to find $y$ . Using Hunter's equation: $y = 2x + 35$ $y = 2(6) + 35$ $y = 12 + 35$ $y = 47$
Verify Solution: Substitute the value of $x$ back into one of the original equations to find $y$ . Using Hunter's equation: $y = 2x + 35$ $y = 2(6) + 35$ $y = 12 + 35$ $y = 47$ Verify the solution by substituting $x$ into the other equation. Using Hunter's partner's equation: $y = x + 41$ $y = 6 + 41$ $y = 47$ Since the value of $y$ is the same in both equations, the solution is correct.