Question

1. $-4x + 9y = 9$

$x - 3y = -6$

1. $-7x + y = -19$

$-2x + 3y = -19$

Explanation:

Problem 10: Solve the system \(

$$\begin{cases}-4x + 9y = 9\\x - 3y = -6\end{cases}$$

\)

Step 1: Solve the second equation for \(x\)

From \(x - 3y = -6\), we can express \(x\) in terms of \(y\) by adding \(3y\) to both sides.
\(x = 3y - 6\)

Step 2: Substitute \(x = 3y - 6\) into the first equation

Substitute \(x\) into \(-4x + 9y = 9\):
\(-4(3y - 6) + 9y = 9\)

Step 3: Simplify and solve for \(y\)

First, distribute the \(-4\):
\(-12y + 24 + 9y = 9\)
Combine like terms:
\(-3y + 24 = 9\)
Subtract 24 from both sides:
\(-3y = 9 - 24\)
\(-3y = -15\)
Divide both sides by \(-3\):
\(y = \frac{-15}{-3} = 5\)

Step 4: Substitute \(y = 5\) back into \(x = 3y - 6\) to find \(x\)

\(x = 3(5) - 6\)
\(x = 15 - 6 = 9\)

Step 1: Solve the first equation for \(y\)

From \(-7x + y = -19\), add \(7x\) to both sides:
\(y = 7x - 19\)

Step 2: Substitute \(y = 7x - 19\) into the second equation

Substitute into \(-2x + 3y = -19\):
\(-2x + 3(7x - 19) = -19\)

Step 3: Simplify and solve for \(x\)

Distribute the 3:
\(-2x + 21x - 57 = -19\)
Combine like terms:
\(19x - 57 = -19\)
Add 57 to both sides:
\(19x = -19 + 57\)
\(19x = 38\)
Divide both sides by 19:
\(x = \frac{38}{19} = 2\)

Step 4: Substitute \(x = 2\) back into \(y = 7x - 19\) to find \(y\)

\(y = 7(2) - 19\)
\(y = 14 - 19 = -5\)

Answer:

\(x = 9\), \(y = 5\)

Problem 12: Solve the system \(

$$\begin{cases}-7x + y = -19\\-2x + 3y = -19\end{cases}$$

\)