Please solve the system with the elimination method. Explain each step as it is performed.
5y = x
2x - 3y = 7
Substitute 5y for x in the second equation.
10y - 3y = 7
7y = 7
y = 1
x = 5y = 5
To solve the system of equations using the elimination method, we need to eliminate one variable by adding or subtracting the equations. Let's start:
Step 1: Multiply the first equation by 2 to make the coefficients of x the same for both equations.
2(5y) = 2(x)
10y = 2x
Step 2: Now we have:
10y = 2x
2x - 3y = 7
Step 3: We want to eliminate the x variable, so we'll multiply the second equation by 5 to make the coefficients of x match.
5(2x - 3y) = 5(7)
10x - 15y = 35
Step 4: Now we have:
10y = 2x
10x - 15y = 35
Step 5: In order to eliminate the y variable, we'll multiply the first equation by -15 to make the coefficients of y match.
-15(10y) = -15(2x)
-150y = -30x
Step 6: Now we have:
-150y = -30x
10x - 15y = 35
Step 7: Now we can add the two equations together to eliminate the y variable.
(-150y) + (-15y) = (-30x) + (10x)
-165y = -20x
Step 8: Finally, divide both sides of the equation by -165 to solve for y.
y = (-20x) / (-165)
y = (4x / 33)
Step 9: Now we substitute the value of y in one of the original equations, let's use the first equation (5y = x).
5(y) = x
5((4x / 33)) = x
(20x / 33) = x
Step 10: Now we can solve for x. To do this, we'll cross-multiply:
33x = 20x
Step 11: Subtract 20x from both sides of the equation:
33x - 20x = 0
13x = 0
Step 12: Finally, divide both sides by 13 to solve for x:
x = 0
Step 13: Now substitute the value of x back into one of the original equations, let's use the first equation (5y = x):
5y = 0
y = 0
Therefore, the solution to the system of equations is x = 0 and y = 0.