Answer
To solve this system of linear equations, we can use the method of substitution.
First, let's solve the first equation for x:
x = 34 - y - z
Now, we substitute this value of x into the second equation:
1(34 - y - z) + 10y + 5z = 100
34 - y - z + 10y + 5z = 100
34 + 9y + 4z = 100
Next, we simplify the second equation:
9y + 4z = 100 - 34
9y + 4z = 66
We can rewrite this equation as:
9y = 66 - 4z
y = (66 - 4z) / 9
Now, we substitute this value of y back into the first equation:
x + (66 - 4z) / 9 + z = 34
Multiplying through by 9 to eliminate the fraction:
9x + 66 - 4z + 9z = 306
9x + 5z = 240
Now we have a system of two equations in two variables:
9x + 5z = 240
9y + 4z = 66
We can solve this using the method of substitution or elimination. Let's use the method of elimination:
Multiplying the first equation by 4 and the second equation by 5, we get:
36x + 20z = 960
45y + 20z = 330
Subtracting the second equation from the first, we eliminate z:
36x - 45y = 630
We can simplify this equation by dividing through by 9:
4x - 5y = 70
Now, let's solve the new system of equations:
4x - 5y = 70
9y + 4z = 66
We can multiply the first equation by 9 and the second equation by 4 to eliminate x:
36x - 45y = 630
36y + 16z = 264
Now, subtracting the first equation from the second, we eliminate y:
36y + 16z - 36x + 45y = 264 - 630
81y + 16z = -366
Dividing through by 3, we get:
27y + 16z = -122
Now, we have a system of two equations in two variables:
4x - 5y = 70
27y + 16z = -122
We can solve this system using the method of substitution or elimination.