Solve using the elimination method.Show work.If the system has no solution or an infinite number of solution state ths...-5x+4y=-16,-15x+2y=-58

To solve the system of equations using the elimination method, we need to eliminate one variable by manipulating the equations.

Given the system of equations:
1) -5x + 4y = -16
2) -15x + 2y = -58

To eliminate one variable, let's multiply equation 1 by 3, and equation 2 by -1, so that the coefficients of y will be the same.

3 * (-5x + 4y) = 3 * (-16)
-1 * (-15x + 2y) = -1 * (-58)

Simplifying, we have:
-15x + 12y = -48
15x - 2y = 58

Now, we can add both equations together to eliminate the x term:
(-15x + 12y) + (15x - 2y) = -48 + 58

Simplifying further, we get:
10y = 10

Dividing both sides by 10, we find:
y = 1

Now, let's substitute y = 1 back into equation 1 to solve for x:
-5x + 4(1) = -16

Simplifying, we have:
-5x + 4 = -16

Subtracting 4 from both sides, we get:
-5x = -20

Dividing both sides by -5, we find:
x = 4

Therefore, the solution to the system of equations is x = 4 and y = 1.

To solve the given system of equations using the elimination method, follow these steps:

Step 1: Multiply one or both equations by a constant to make the coefficients of either x or y the same (opposite in sign) for both equations. In this case, we need to eliminate one variable by making the coefficients of either x or y in both equations the same.

Let's start by making the coefficients of x the same in both equations. To do this, we'll multiply the first equation by -3 and the second equation by -5:
Equation 1: -3(-5x + 4y) = -3(-16) -> 15x - 12y = 48
Equation 2: -5(-15x + 2y) = -5(-58) -> 75x - 10y = 290

Now we have two new equations:
15x - 12y = 48 ..... (Equation 3)
75x - 10y = 290 ..... (Equation 4)

Step 2: Either add or subtract the new equations to eliminate one variable. We'll eliminate y by subtracting Equation 3 from Equation 4:

(Equation 4 - Equation 3):
(75x - 10y) - (15x - 12y) = 290 - 48
75x - 10y - 15x + 12y = 242

Combine like terms:
(75x - 15x) + (-10y + 12y) = 242
60x + 2y = 242 ..... (Equation 5)

Step 3: Solve the resulting equation for one variable. Let's solve Equation 5 for y:
2y = 242 - 60x
y = (242 - 60x) / 2
y = 121 - 30x ..... (Equation 6)

Step 4: Substitute the value of y from Equation 6 into either Equation 3 or Equation 4 to solve for x. Let's substitute y in Equation 3:

15x - 12(121 - 30x) = 48
15x - 1452 + 360x = 48
Combine like terms:
375x - 1452 = 48

Step 5: Solve the equation obtained in Step 4 for x:
375x = 48 + 1452
375x = 1500
x = 1500 / 375
x = 4

Step 6: Substitute the value of x back into Equation 6 to find the value of y:
y = 121 - 30(4)
y = 121 - 120
y = 1

Therefore, the solution to the system of equations -5x + 4y = -16 and -15x + 2y = -58 is x = 4 and y = 1.