Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Jake and his workout partner are lifting weights together, doing many sets of each exercise. On a certain exercise, Jake is using a $17$ -kilogram bar, increasing the amount of weight he lifts by $3$ kilograms on each set. His partner, meanwhile, started out using an $11$ -kilogram bar and is upping the weight by adding $9$ kilograms on every set. Eventually, Jake 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 $\_$ kilograms.

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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

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

17

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

3

kilograms on each set. His partner, meanwhile, started out using an

11

-kilogram bar and is upping the weight by adding

9

kilograms on every set. Eventually, Jake 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

\_

kilograms.

Define Variables: Let's define the variables: $\newline$ Let $x$ represent the number of sets completed. $\newline$ Let $y$ represent the total weight lifted by Jake or his partner after $x$ sets. $\newline$ Jake's starting weight is $17$ kilograms and he adds $3$ kilograms per set. So, the equation for Jake's total weight lifted after $x$ sets is: $\newline$ $y = 3x + 17$
Jake's Weight Equation: Jake's partner starts with $11$ kilograms and adds $9$ kilograms per set. The equation for his partner's total weight lifted after $x$ sets is: $\newline$ $y = 9x + 11$
Partner's Weight Equation: Now we have a system of two equations: $\newline$ $1$ ) $y = 3x + 17$ (Jake's weight) $\newline$ $2$ ) $y = 9x + 11$ (Partner's weight) $\newline$ To find the point where they will be lifting the same amount, we set the two equations equal to each other: $\newline$ $3x + 17 = 9x + 11$
Set Equations Equal: We solve for $x$ by subtracting $3x$ from both sides: $\newline$ $3x + 17 - 3x = 9x + 11 - 3x$ $\newline$ $17 = 6x + 11$
Solve for x: Next, we subtract $11$ from both sides to isolate the term with $x$ : $\newline$ $17 - 11 = 6x + 11 - 11$ $\newline$ $6 = 6x$
Isolate x Term: Now we divide both sides by $6$ to solve for x: $\newline$ $\frac{6}{6} = \frac{6x}{6}$ $\newline$ $1 = x$
Divide to Solve $x$ : We have found that $x = 1$ , which means they will be lifting the same amount of weight after completing $1$ set. Now we need to find out how much weight that is. We can substitute $x$ back into either of the original equations. Let's use Jake's equation: $\newline$ $y = 3x + 17$ $\newline$ $y = 3(1) + 17$ $\newline$ $y = 3 + 17$ $\newline$ $y = 20$
Calculate Total Weight: So, after completing $1$ set, they will both be lifting $20$ kilograms.