Question

Any help on this ? 30 pts

Answer 1

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:

$\sf x+3x+7y=41$

Combine like terms(x + 3x = 4x):

$\sf 4x+7y=41$

We are also told that the perimeter of the rectangle is 66 inches. The formula for the perimeter of a rectangle is:

$\sf P=2(l+w)$

Plug in what we know:

$\sf 66=2(3x+7y)$

Distribute 2 into the parenthesis:

$\sf 6x+14y=66$

So our system of equations is:

$\sf 4x+7y=41$
$\sf 6x+14y=66$

Let's use the substitution method to solve this system. Let's solve for 'x' in the second equation:

$\sf 6x+14y=66$

Subtract 14y to both sides:

$\sf 6x=-14y+66$

Divide 6 to both sides:

$\sf x=-\dfrac{7}{3}y+11$

Now let's plug this in for 'x' in the first equation:

$\sf 4x+7y=41$

$\sf 4(-\dfrac{7}{3}y+11)+7y=41$

Distribute:

$\sf -\dfrac{28}{3}y+44+7y=41$

Combine like terms:

$\sf -\dfrac{7}{3}y+44=41$

Subtract 44 to both sides:

$\sf -\dfrac{7}{3}y=-3$

Divide -7/3 to both sides or multiply by its reciprocal -3/7:

$\sf y=\dfrac{9}{7}$

So the solution to the system of equations is:

$\sf (8,\dfrac{9}{7})$

Plug these values in to find the length and width of the rectangle.

Length:

$\sf 3x$

$\sf 3(8)$

$\boxed{\sf 24}$

Width:

$\sf 7y$

$\sf 7(\dfrac{9}{7})$

$\boxed{\sf 9}$