Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.

10x + 5y = 52.5

30x – 7y = 80.5

To solve the given system of equations using the elimination method, we'll manipulate the equations by multiplying them so that the coefficients of either the x-term or y-term will become equal in magnitude but opposite in sign. We'll start by eliminating the y-term.

Multiply the first equation by 7 and the second equation by 5 to make the coefficients of the y-term in both equations equal:

7(10x + 5y) = 7(52.5)
5(30x – 7y) = 5(80.5)

This simplifies the equations to:

70x + 35y = 367.5
150x - 35y = 402.5

Now, add the two equations together to eliminate the y-term:

(70x + 35y) + (150x - 35y) = 367.5 + 402.5

Combine like terms:

220x = 770

Divide both sides of the equation by 220:

x = 770 / 220
x = 3.5

Now that we have the value of x, substitute it back into one of the original equations to solve for y. Let's use the first equation:

10x + 5y = 52.5

Substitute x = 3.5:

10(3.5) + 5y = 52.5
35 + 5y = 52.5

Subtract 35 from both sides:

5y = 52.5 - 35
5y = 17.5

Divide both sides by 5:

y = 17.5 / 5
y = 3.5

Therefore, the solution to the system of equations is x = 3.5 and y = 3.5.

Multiply the first equation by 3, then subtract the second from it. Solve for y. Then insert the y value into one of the equations to find x. Then make your statement.