Let's work through this step-by-step to help fill in the blanks.

Step 1

We have the equation: $x - 3y = -8$ Solving for $x$: $x = -8 + 3y$

Step 2

Now, substitute $x = -8 + 3y$ into Equation 1 and Equation 3.

Substitute into Equation 1:

Original Equation 1: $2x - 4y + 5z = -30$ Substituting $x = -8 + 3y$: $2(-8 + 3y) - 4y + 5z = -30$ Expanding this: $-16 + 6y - 4y + 5z = -30$ Combine terms: $2y + 5z = -14$

Substitute into Equation 3:

Original Equation 3: $-2x + 2y - 5z = 24$ Substituting $x = -8 + 3y$: $-2(-8 + 3y) + 2y - 5z = 24$ Expanding this: $16 - 6y + 2y - 5z = 24$ Combine terms: $-4y - 5z = 8$

So, the equations we have after substitution are:

• Equation 1: $2y + 5z = -14$
• Equation 3: $-4y - 5z = 8$

Step 3

Rewrite these two equations as follows:

• Equation 1: $2y + 5z = -14$
• Equation 3: $-4y - 5z = 8$

Step 4

Now, add the two equations to eliminate $z$ and solve for $y$: $(2y + 5z) + (-4y - 5z) = -14 + 8$ This simplifies to: $-2y = -6$ Solving for $y$: $y = 3$

With $y = 3$, we can substitute back to find $x$ and $z$ in further steps. Let me know if you want to continue or have questions on any of the above steps!

Related Questions

1. How do you solve for $z$ once $y$ is found?
2. What happens if there is no unique solution to this system?
3. How can substitution be used in different types of systems of equations?
4. Why is it useful to eliminate variables when solving systems of equations?
5. What are alternative methods to solve this system besides substitution?

Tip

When solving systems of equations, carefully combine like terms to avoid errors, especially with signs.