Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ There are two trails near Sandra's house that she runs regularly, a short loop and a long loop. Last week, she ran $6$ short loops and $5$ long loops, for a total of $38$ miles. This week, she ran $6$ short loops and $3$ long loops, covering a total of $30$ miles. What is the length of each loop? $\newline$ The short loop has a length of $\_$ miles, and the long loop has a length of $\_$ miles.

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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

There are two trails near Sandra's house that she runs regularly, a short loop and a long loop. Last week, she ran

6

short loops and

5

long loops, for a total of

38

miles. This week, she ran

6

short loops and

3

long loops, covering a total of

30

miles. What is the length of each loop?

\newline

The short loop has a length of

\_

miles, and the long loop has a length of

\_

miles.

Define variables: Define the variables for the lengths of the short and long loops. $\newline$ Let $x$ be the length of the short loop in miles. $\newline$ Let $y$ be the length of the long loop in miles.
Write equations: Write the system of equations based on the given information. $\newline$ First week: $6$ short loops and $5$ long loops for a total of $38$ miles. $\newline$ $6x + 5y = 38$ $\newline$ Second week: $6$ short loops and $3$ long loops for a total of $30$ miles. $\newline$ $6x + 3y = 30$
Eliminate variable: Decide which variable to eliminate. $\newline$ We can eliminate $x$ by subtracting the second equation from the first because the coefficients of $x$ are the same in both equations.
Subtract equations: Subtract the second equation from the first to eliminate $x$ . $\newline$ $(6x + 5y) - (6x + 3y) = 38 - 30$ $\newline$ $6x + 5y - 6x - 3y = 38 - 30$ $\newline$ $2y = 8$
Solve for y: Solve for y. $\newline$ $2y = 8$ $\newline$ $y = \frac{8}{2}$ $\newline$ $y = 4$
Substitute value: Substitute the value of $y$ back into one of the original equations to solve for $x$ . Using the second equation: $6x + 3(4) = 30$ $6x + 12 = 30$ $6x = 30 - 12$ $6x = 18$ $x = 18 / 6$ $x = 3$
Verify solution: Verify the solution by substituting the values of $x$ and $y$ into the other equation. $\newline$ Using the first equation: $6(3) + 5(4) = 38$ $\newline$ $18 + 20 = 38$ $\newline$ $38 = 38$ $\newline$ The values satisfy the equation, so the solution is correct.