Certainly! Let's go through this step by step to find the length of each remaining side.
First, let's use the information given in the problem:
1. We know the perimeter of the lot is given by the expression [tex]\( 13y + 5 \)[/tex].
2. The lengths of three sides are given by the expressions: [tex]\( 4y + 8 \)[/tex], [tex]\( 2y + 4 \)[/tex], and [tex]\( 2y - 1 \)[/tex].
3. The remaining two sides are of equal length, let's denote each of these side lengths as [tex]\( a \)[/tex].
Since the perimeter is the sum of all side lengths, we can set up the following equation:
[tex]\[ 13y + 5 = (4y + 8) + (2y + 4) + (2y - 1) + a + a \][/tex]
Let's simplify this equation step by step:
1. Combine the known side lengths:
[tex]\[ (4y + 8) + (2y + 4) + (2y - 1) = 4y + 8 + 2y + 4 + 2y - 1 \][/tex]
2. Combine like terms on the right side:
[tex]\[ 4y + 2y + 2y + 8 + 4 - 1 = 8y + 11 \][/tex]
The equation now looks like this:
[tex]\[ 13y + 5 = 8y + 11 + 2a \][/tex]
3. To isolate [tex]\( 2a \)[/tex], we'll subtract [tex]\( 8y + 11 \)[/tex] from both sides:
[tex]\[ 13y + 5 - 8y - 11 = 2a \][/tex]
4. Simplify the left side:
[tex]\[ 5y - 6 = 2a \][/tex]
5. Now, to find [tex]\( a \)[/tex], divide both sides by 2:
[tex]\[ a = \frac{5y - 6}{2} \][/tex]
So, the expression that gives the length of each remaining side is:
[tex]\[ \boxed{\frac{5y - 6}{2}} \][/tex]
For example, if [tex]\( y = 2 \)[/tex], the lengths of all sides would be:
- Perimeter expression: [tex]\( 13y + 5 = 31 \)[/tex]
- Side 1: [tex]\( 4y + 8 = 16 \)[/tex]
- Side 2: [tex]\( 2y + 4 = 8 \)[/tex]
- Side 3: [tex]\( 2y - 1 = 3 \)[/tex]
- Each of the remaining two sides: [tex]\( \frac{5y - 6}{2} = 2.0 \)[/tex]
Hence, the lengths of the remaining sides when [tex]\( y = 2 \)[/tex] are 2.0, confirming the calculation:
[tex]\[ \boxed{\frac{5y - 6}{2}} \][/tex] as the correct expression for each remaining side.
