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.