To solve the system of equations using substitution, here are the equations from the image:

1. $8x + 3y = 4$
2. $-7x + 5y = -34$

Step 1: Solve one equation for one variable

From the first equation ($8x + 3y = 4$), solve for $y$ in terms of $x$:

$3y = 4 - 8x$ $y = \frac{4 - 8x}{3}$

Step 2: Substitute $y$ into the second equation

Substitute $y = \frac{4 - 8x}{3}$ into the second equation ($-7x + 5y = -34$):

$-7x + 5\left(\frac{4 - 8x}{3}\right) = -34$

Simplify the terms: $-7x + \frac{5(4 - 8x)}{3} = -34$ $-7x + \frac{20 - 40x}{3} = -34$

Combine the terms under a common denominator: $\frac{-21x}{3} + \frac{20 - 40x}{3} = -34$ $\frac{-21x + 20 - 40x}{3} = -34$

Combine like terms: $\frac{-61x + 20}{3} = -34$

Step 3: Solve for $x$

Multiply through by 3 to eliminate the fraction: $-61x + 20 = -102$

Subtract 20 from both sides: $-61x = -122$

Divide by $-61$: $x = 2$

Step 4: Solve for $y$

Substitute $x = 2$ back into $y = \frac{4 - 8x}{3}$: $y = \frac{4 - 8(2)}{3}$ $y = \frac{4 - 16}{3}$ $y = \frac{-12}{3}$ $y = -4$

Final Solution:

The solution to the system of equations is: $x = 2, \; y = -4$

Would you like a step-by-step verification or have any questions about this process?

Here are 5 related questions to expand your understanding:

1. What are other methods to solve a system of linear equations (e.g., elimination)?
2. How can you verify the solution to a system of equations?
3. What happens if a system of equations has no solution or infinitely many solutions?
4. How can you graphically interpret the solution to this system of equations?
5. How would you solve this system if decimals or fractions appeared in the original equations?

Tip: Always double-check your solution by substituting it back into both original equations!