Questions: Tamara has 140 feet of fencing to construct a rectangular area next to a house. Tamara will only need to fence three sides, because the fourth side will be the wall of the house. Lastly, Tamara wants the length of the enclosure (parallel to the house wall) to be 20 feet more than three times the width. Using the variables l and w to represent the length and width of the enclosure, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Tamara has 140 feet of fencing to construct a rectangular area next to a house. Tamara will only need to fence three sides, because the fourth side will be the wall of the house. Lastly, Tamara wants the length of the enclosure (parallel to the house wall) to be 20 feet more than three times the width. Using the variables l and w to represent the length and width of the enclosure, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Transcript text: Tamara has 140 feet of fencing to construct a rectangular area next to a house. Tamara will only need to fence three sides, because the fourth side will be the wall of the house. Lastly, Tamara wants the length of the enclosure (parallel to the house wall) to be 20 feet more than three times the width. Using the variables $l$ and $w$ to represent the length and width of the enclosure, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Solution

Solution Steps

To solve this problem, we need to set up a system of equations based on the given conditions. The first equation comes from the perimeter constraint: since only three sides are fenced, the equation is $ l + 2w = 140 $. The second equation is derived from the relationship between length and width: $ l = 3w + 20 $. We can solve this system of equations to find the values of $ l $ and $ w $.

Step 1: Set Up the System of Equations

We are given that Tamara has 140 feet of fencing to construct a rectangular area next to a house, fencing only three sides. The length $ l $ of the enclosure is 20 feet more than three times the width $ w $. We can express these conditions with the following system of equations:

$ l + 2w = 140 $
$ l = 3w + 20 $

Step 2: Solve the System of Equations

Substitute the expression for $ l $ from the second equation into the first equation: \[ 3w + 20 + 2w = 140 \] Simplify and solve for $ w $: \[ 5w + 20 = 140 \\ 5w = 120 \\ w = 24 \]

Step 3: Find the Length

Substitute $ w = 24 $ back into the equation for $ l $: \[ l = 3(24) + 20 \\ l = 72 + 20 \\ l = 92 \]