19.03.2023 - 13:34

# Solve each system by substitution. 1) -x – y – 3z = -9 z = -3x – 1 x = 5y – z + 23 2) x = -4z – 19 y = 5x + z – 4

Solve each system by substitution.

1) -x – y – 3z = -9

z = -3x – 1

x = 5y – z + 23

2) x = -4z – 19

y = 5x + z – 4

-5y – z = 25

3) y = x + z + 5

z = -3y – 3

2x – y = -4

• April 4, 2023 в 08:59

Sure, I'll explain how to solve each system by substitution.

1. The second equation gives us z in terms of x: z = -3x - 1. We can substitute this expression for z in the first equation, which gives us: -x - y - 3(-3x - 1) = -9. Simplifying this equation, we get -10x - y = -6. We can now substitute the expression for x from the third equation, x = 5y - z + 23, into this equation: -10(5y - z + 23) - y = -6. Simplifying this equation, we get -51y + 10z = 276. Finally, we substitute the expression for z from the second equation, z = -3x - 1, into this equation: -51y + 10(-3x - 1) = 276. Simplifying this equation, we get -51y - 30x = 286. This is a linear system of two equations with two variables. We can solve for y in the first equation and substitute into the second equation to find x.

2. The first equation gives us x in terms of z: x = -4z - 19. We can substitute this expression for x in the second equation, which gives us: y = 5(-4z - 19) + z - 4. Simplifying this equation, we get y = -20z - 99. We can now substitute these expressions for x and y in the third equation, which gives us: -5(-20z - 99) - z = 25. Simplifying this equation, we get z = -4. We can now substitute this value of z into the expressions we found for x and y, to get x = 13 and y = 1. This is the solution of the system.

3. The second equation gives us z in terms of y: z = -3y - 3. We can substitute this expression for z in the first equation, which gives us: y = x + (-3y - 3) + 5. Simplifying this equation, we get 4y + x = 8. We can now substitute the expression for y from this equation into the third equation, which gives us: 2x - (4y + x) = -4. Simplifying this equation, we get x = 2. We can now substitute this value of x into the equation we found for y, 4y + x = 8, to get y = 1. Finally, we can substitute the values of x and y into the expression we found for z, z = -3y - 3, to get z = -6. This is the solution of the system.