To solve the system of equations by elimination, we will first eliminate the variable ×.
The system of equations is:
1. × + 5y - z = -10
2. 2× - y + 5z + 5z = -9
3. 2× - 10y - 5z = 0
We can solve the system of equations in the following steps:
Equation 2 can be simplified by adding the two terms with z:
2× - y + 10z = -9
Now we have two equations with the same value of ×:
× + 5y - z = -10
2× - y + 10z = -9
To eliminate ×, we need to multiply equation 1 by -2:
-2(× + 5y - z) = -2(-10)
-2× - 10y + 2z = 20
Now we have:
-2× - 10y + 2z = 20
2× - y + 10z = -9
Adding the equations together, the × variable is eliminated:
(-2× + 2×) + (-10y - y) + (2z + 10z) = 20 - 9
-11y + 12z = 11
Now we have the simplified equation:
-11y + 12z = 11 ...(4)
Next, we need to eliminate the variable × again in the third equation:
2× - 10y - 5z = 0
Multiplying the first equation by 2, we get:
2(× + 5y - z) = 2(-10)
2× + 10y - 2z = -20
Now we have:
2× + 10y - 2z = -20
2× - 10y - 5z = 0
Adding the equations together, the × variable is eliminated:
(2× + 2×) + (10y - 10y) + (-2z - 5z) = -20 + 0
0 + (-7z) = -20
-7z = -20
Simplifying the equation, we find:
z = -20 / -7
z = 20/7 ...(5)
Substituting the value of z in equation 4, we find:
-11y + 12(20/7) = 11
-11y + 240/7 = 11
Multiplying the equation by 7 to eliminate the fraction, we get:
-77y + 240 = 77
-77y = 77 - 240
-77y = -163
y = -163 / -77
y = 163/77 ...(6)
Finally, substituting the values of y and z in equation 1, we find:
x + 5(163/77) - 20/7 = -10
Multiplying the equation by 77 to eliminate the fraction, we get:
77x + 5(163) - 20(11) = -770
77x + 815 - 220 = -770
77x = -770 - 815 + 220
77x = -1365
x = -1365 / 77
x = -45/11 ...(7)
Therefore, the solution to the system of elimination is:
x = -45/11, y = 163/77, z = 20/7.
solve the system of elimination. {×+5y-z=-10; 2×-y+5z+5z=-9; 2×-10y-5z=0
1 answer