Question

1. \\( y = -x + 4 \\)\\( 5x + 6y = 13 \\)\\( \checkmark \\)check:\
2. \\( -3x + 4y = 7 \\)\\( y = 3x - 5 \\)\\( \checkmark \\)check:

Explanation:

Problem 3: Solve the system \(

$$\begin{cases} y = -x + 4 \\ 5x + 6y = 13 \end{cases}$$

\) and check.

Step 1: Substitute \( y = -x + 4 \) into \( 5x + 6y = 13 \)

Substitute \( y \) in the second equation with \( -x + 4 \) from the first equation.
\( 5x + 6(-x + 4) = 13 \)

Step 2: Simplify and solve for \( x \)

Expand the left side: \( 5x - 6x + 24 = 13 \)
Combine like terms: \( -x + 24 = 13 \)
Subtract 24 from both sides: \( -x = 13 - 24 \)
\( -x = -11 \)
Multiply both sides by -1: \( x = 11 \)

Step 3: Find \( y \) using \( y = -x + 4 \)

Substitute \( x = 11 \) into \( y = -x + 4 \):
\( y = -11 + 4 \)
\( y = -7 \)

Step 4: Check the solution in both equations

Check in \( y = -x + 4 \):

Left side: \( y = -7 \)
Right side: \( -11 + 4 = -7 \)
Left side = Right side.

Check in \( 5x + 6y = 13 \):

Left side: \( 5(11) + 6(-7) = 55 - 42 = 13 \)
Right side: \( 13 \)
Left side = Right side.

Problem 4: Solve the system \(

$$\begin{cases} -3x + 4y = 7 \\ y = 3x - 5 \end{cases}$$

\) and check.

Step 1: Substitute \( y = 3x - 5 \) into \( -3x + 4y = 7 \)

Substitute \( y \) in the first equation with \( 3x - 5 \) from the second equation.
\( -3x + 4(3x - 5) = 7 \)

Step 2: Simplify and solve for \( x \)

Expand the left side: \( -3x + 12x - 20 = 7 \)
Combine like terms: \( 9x - 20 = 7 \)
Add 20 to both sides: \( 9x = 7 + 20 \)
\( 9x = 27 \)
Divide both sides by 9: \( x = 3 \)

Step 3: Find \( y \) using \( y = 3x - 5 \)

Substitute \( x = 3 \) into \( y = 3x - 5 \):
\( y = 3(3) - 5 \)
\( y = 9 - 5 \)
\( y = 4 \)

Step 4: Check the solution in both equations

Check in \( y = 3x - 5 \):

Left side: \( y = 4 \)
Right side: \( 3(3) - 5 = 9 - 5 = 4 \)
Left side = Right side.

Check in \( -3x + 4y = 7 \):

Left side: \( -3(3) + 4(4) = -9 + 16 = 7 \)
Right side: \( 7 \)
Left side = Right side.

Answer:

s:

Problem 3:

Solution: \( x = 11 \), \( y = -7 \)

Problem 4:

Solution: \( x = 3 \), \( y = 4 \)