Final answer:
The solution to the first system of equations is (2, 0). The sum of the solutions to the second system of equations is 8.
Step-by-step explanation:
To find the solution to the system of linear equations, we can use the method of substitution. First, solve one of the equations for one variable in terms of the other. Let's solve the first equation for y:
2x - y = 4 ⟶ y = 2x - 4
Next, substitute this expression for y into the second equation:
5x + 2(2x - 4) = 10
Simplify and solve for x:
5x + 4x - 8 = 10 ⟶ 9x = 18 ⟶ x = 2
Now, substitute this value of x back into the first equation to find y:
y = 2(2) - 4 ⟶ y = 0
Therefore, the solution to the system of equations is (2, 0).
To find the sum of the three solutions for the second system of equations, we need to solve it first. Using the method of substitution again:
From the first equation, x = 5 - 3y + 2z
Substitute this expression for x into the second equation:
(5 - 3y + 2z) - 3y + 2z = -6
Simplify and solve for y:
-6y + 4z = 11
To solve the third equation, substitute the values of x and y into the equation:
3(5 - 3y + 2z) + y - 4z = -8
After simplifying, we get:
-9y + 10z = -23
Solving the two equations simultaneously gives y = 1 and z = 3.
So, the three solutions are x = 4, y = 1, z = 3. The sum of these values is 4 + 1 + 3 = 8.