Solve the Linear Equations: -4x + 4y = 15 and 2x + 8y = 12 Using Substitution

To solve this problem, we'll apply the substitution method, following these steps:

• Step 1: Solve the first equation for one of the variables.
• Step 2: Substitute this expression into the second equation and solve for the other variable.
• Step 3: Use the value found in Step 2 in the rearranged first equation to find the remaining variable.

Step-by-Step Solution:

Step 1: By using the first equation, $-4x + 4y = 15$, we can solve for $y$.

Step 1.1: Simplify the equation to solve for $y$ by adding $4x$ to both sides:
$4y = 4x + 15$

Step 1.2: Divide every term by 4:
$y = x + \frac{15}{4}$

Step 2: Substitute the expression for $y$ into the second equation, $2x + 8y = 12$.

Step 2.1: Substitute $y = x + \frac{15}{4}$:
$2x + 8(x + \frac{15}{4}) = 12$

Step 2.2: Simplify and solve for $x$:
$2x + 8x + 30 = 12$

Combine like terms:
$10x + 30 = 12$

Subtract 30 from both sides:
$10x = 12 - 30$

Resulting in:
$10x = -18$

Divide by 10:
$x = -\frac{9}{5}$

Step 3: Substitute $x = -\frac{9}{5}$ back into the expression for $y$:

$y = -\frac{9}{5} + \frac{15}{4}$

Convert fractions to a common denominator, which is 20:
$y = -\frac{36}{20} + \frac{75}{20}$

Solve by combining terms:
$y = \frac{39}{20}$

Thus, the solution to the system is $x = -\frac{9}{5}$ and $y = \frac{39}{20}$.

Therefore, the correct solution is identified as choice 4.