The perimeter is adding up all sides of a shape. Here we are told that the perimeter of the triangle is 41 inches. So we can make the equation:
[tex]\sf x+3x+7y=41[/tex]
Combine like terms(x + 3x = 4x):
[tex]\sf 4x+7y=41[/tex]
We are also told that the perimeter of the rectangle is 66 inches. The formula for the perimeter of a rectangle is:
[tex]\sf P=2(l+w)[/tex]
Plug in what we know:
[tex]\sf 66=2(3x+7y)[/tex]
Distribute 2 into the parenthesis:
[tex]\sf 6x+14y=66[/tex]
So our system of equations is:
[tex]\sf 4x+7y=41[/tex]
[tex]\sf 6x+14y=66[/tex]
Let's use the substitution method to solve this system. Let's solve for 'x' in the second equation:
[tex]\sf 6x+14y=66[/tex]
Subtract 14y to both sides:
[tex]\sf 6x=-14y+66[/tex]
Divide 6 to both sides:
[tex]\sf x=-\dfrac{7}{3}y+11[/tex]
Now let's plug this in for 'x' in the first equation:
[tex]\sf 4x+7y=41[/tex]
[tex]\sf 4(-\dfrac{7}{3}y+11)+7y=41[/tex]
Distribute:
[tex]\sf -\dfrac{28}{3}y+44+7y=41[/tex]
Combine like terms:
[tex]\sf -\dfrac{7}{3}y+44=41[/tex]
Subtract 44 to both sides:
[tex]\sf -\dfrac{7}{3}y=-3[/tex]
Divide -7/3 to both sides or multiply by its reciprocal -3/7:
[tex]\sf y=\dfrac{9}{7}[/tex]
So the solution to the system of equations is:
[tex]\sf (8,\dfrac{9}{7})[/tex]
Plug these values in to find the length and width of the rectangle.
Length:
[tex]\sf 3x[/tex]
[tex]\sf 3(8)[/tex]
[tex]\boxed{\sf 24}[/tex]
Width:
[tex]\sf 7y[/tex]
[tex]\sf 7(\dfrac{9}{7})[/tex]
[tex]\boxed{\sf 9}[/tex]
