Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Hannah practiced her beam routine $1$ time and her floor routine $7$ times, which took a total of $15$ minutes. During afternoon practice, she ran through her her beam routine $2$ times and her floor routine $2$ times, which took a total of $6$ minutes. How long is each routine? $\newline$ The beam routine is $\_$ minutes long and the floor exercise is $\_$ minutes long.

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Algebra 1

Solve a system of equations using elimination: word problems

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

\newline

Gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Hannah practiced her beam routine

1

time and her floor routine

7

times, which took a total of

15

minutes. During afternoon practice, she ran through her her beam routine

2

times and her floor routine

2

times, which took a total of

6

minutes. How long is each routine?

\newline

The beam routine is

\_

minutes long and the floor exercise is

\_

minutes long.

Define variables: Define the variables for the routines. $\newline$ Let $x$ be the time for the beam routine and $y$ be the time for the floor routine.
Write equations: Write the equations based on the given information. $\newline$ For the morning practice, we have: $\newline$ $1x + 7y = 15$ $\newline$ For the afternoon practice, we have: $\newline$ $2x + 2y = 6$
Set up system: Set up the system of equations. $\newline$ We have the following system: $\newline$ $x + 7y = 15$ $\newline$ $2x + 2y = 6$
Eliminate variable: Decide which variable to eliminate. $\newline$ We can eliminate $x$ by multiplying the first equation by $-2$ and adding it to the second equation.
Perform addition: Multiply the first equation by $-2$ and add it to the second equation to eliminate $x$ .
$-2(x + 7y) = -2(15)$
$-2x - 14y = -30$
Now add this to the second equation:
$(2x + 2y) + (-2x - 14y) = 6 + (-30)$
Simplify equation: Perform the addition to eliminate $x$ . $2x - 2x + 2y - 14y = 6 - 30$ $0x - 12y = -24$ Simplify the equation: $-12y = -24$
Solve for y: Solve for y. $\newline$ Divide both sides by $-12$ : $\newline$ $y = \frac{-24}{-12}$ $\newline$ $y = 2$
Substitute and solve: Substitute $y$ back into one of the original equations to solve for $x$ . Using the first equation: $x + 7(2) = 15$ $x + 14 = 15$
Final solution: Solve for $x$ . $x + 14 - 14 = 15 - 14$ $x = 1$