Question

1. $y = -3x$

$x - y = 4$

1. $y = -\frac{3}{5}x + 2$

$y = \frac{1}{5}x - 2$
ed qu it
$(-4, 0)$ $(1, -3)$ $(1, -1)$

Explanation:

Problem 4: Solve the system \( y = -3x \) and \( x - y = 4 \) and verify the solution \((1, -3)\)

Step 1: Substitute \( y = -3x \) into \( x - y = 4 \)

Substitute \( y \) in the second equation with \( -3x \) from the first equation. So we get \( x - (-3x) = 4 \). Simplifying the left side: \( x + 3x = 4 \), which is \( 4x = 4 \).

Step 2: Solve for \( x \)

Divide both sides of \( 4x = 4 \) by 4: \( x = \frac{4}{4} = 1 \).

Step 3: Solve for \( y \)

Substitute \( x = 1 \) into \( y = -3x \). So \( y = -3(1) = -3 \).

Step 4: Verify the solution \((1, -3)\) in both equations

• For \( y = -3x \): Left side \( y = -3 \), Right side \( -3x = -3(1) = -3 \). So \( -3 = -3 \), true.
• For \( x - y = 4 \): Left side \( x - y = 1 - (-3) = 1 + 3 = 4 \), Right side \( 4 \). So \( 4 = 4 \), true.

Problem 8: Solve the system \( y = -\frac{3}{5}x + 2 \) and \( y = \frac{1}{5}x - 2 \) and verify the solution \((5, -1)\)

Step 1: Set the two equations equal to each other (since both equal \( y \))

\( -\frac{3}{5}x + 2 = \frac{1}{5}x - 2 \)

Step 2: Solve for \( x \)

Add \( \frac{3}{5}x \) to both sides: \( 2 = \frac{1}{5}x + \frac{3}{5}x - 2 \). Combine like terms: \( 2 = \frac{4}{5}x - 2 \). Add 2 to both sides: \( 4 = \frac{4}{5}x \). Multiply both sides by \( \frac{5}{4} \): \( x = 4\times\frac{5}{4} = 5 \).

Step 3: Solve for \( y \)

Substitute \( x = 5 \) into \( y = \frac{1}{5}x - 2 \) (we could use either equation). \( y = \frac{1}{5}(5) - 2 = 1 - 2 = -1 \).

Step 4: Verify the solution \((5, -1)\) in both equations

• For \( y = -\frac{3}{5}x + 2 \): Left side \( y = -1 \), Right side \( -\frac{3}{5}(5) + 2 = -3 + 2 = -1 \). So \( -1 = -1 \), true.
• For \( y = \frac{1}{5}x - 2 \): Left side \( y = -1 \), Right side \( \frac{1}{5}(5) - 2 = 1 - 2 = -1 \). So \( -1 = -1 \), true.

Answer:

s:

• For problem 4: The solution of the system is \(\boldsymbol{(1, -3)}\) (verified).
• For problem 8: The solution of the system is \(\boldsymbol{(5, -1)}\) (verified).