Find the solution of the system of equations.
x, minus, 10, y, equals, 18
x−10y=
18
minus, 6, x, minus, 10, y, equals, 32
−6x−10y=
32
To find the solution of the system of equations, we can use the method of substitution or elimination.
Method of substitution:
1. Solve one equation for a variable in terms of the other variable.
From the first equation, we can solve for x:
x - 10y = 18
x = 10y + 18
2. Substitute the expression obtained in step 1 into the other equation.
Substituting x = 10y + 18 into the second equation:
-6x - 10y = 32
-6(10y + 18) - 10y = 32
-60y - 108 - 10y = 32
-70y - 108 = 32
-70y = 32 + 108
-70y = 140
y = 140/(-70)
y = -2
3. Substitute the value of y back into one of the original equations to find the value of x.
Using the first equation:
x - 10y = 18
x - 10(-2) = 18
x + 20 = 18
x = 18 - 20
x = -2
Therefore, the solution to the system of equations is x = -2 and y = -2.
Method of elimination:
1. Multiply one or both equations by a constant to make the coefficients of one of the variables opposite in sign.
Let's multiply the first equation by 6 and the second equation by -1 to make the coefficients of y opposite:
6(x - 10y) = 6(18)
-(-6x - 10y) = -(32)
Expanding:
6x - 60y = 108
6x + 10y = 32
2. Add the two equations together to eliminate the x term:
(6x - 60y) + (6x + 10y) = 108 + 32
12x - 50y = 140
3. Solve this new equation for one of the variables.
Let's solve for x:
12x - 50y = 140
12x = 50y + 140
x = (50y + 140)/12
4. Substitute this expression for x back into one of the original equations to solve for the other variable.
Using the first equation:
x - 10y = 18
(50y + 140)/12 - 10y = 18
(50y + 140) - 120y = 216
-70y + 140 = 216
-70y = 76
y = 76/(-70)
y = -2
5. Substitute the value of y back into the equation from step 3 to find the value of x:
x = (50y + 140)/12
x = (50(-2) + 140)/12
x = (-100 + 140)/12
x = 40/12
x = 10/3
So the solution to the system of equations is x = 10/3 and y = -2, which is the same result obtained using the method of substitution.