There are two numbers, that sum up to 53. Three times the smaller number is equal to 19 more than the larger number. What are the numbers ?

x = 18  (Smaller number)

y = 35 (Larger number)

Step-by-step explanation:

To solve, begin by setting up a system of equations.

'x' being the smaller number, while

'y' is the larger number.

We can construct the equations:

x + y = 53

3x = y + 19

Rearrange the second equation to equal 'y':

3x - 19 = y

Plug this value of 'y' into the first equation:

x + 3x - 19 = 53

Combine like terms and simplify:

4x - 19 = 53

4x = 72

x = 18

Plug in this value for 'x' into an equation to solve for 'y':

18 + y = 53

y = 35.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240