Question

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Explanation:

Problem 3: Solve the system \(

$$\begin{cases} x + y = 30 \\ x - y = 6 \end{cases}$$

\)

Step 1: Add the two equations

Add the equations \( x + y = 30 \) and \( x - y = 6 \) to eliminate \( y \).
\( (x + y) + (x - y) = 30 + 6 \)
Simplify the left side: \( x + y + x - y = 2x \)
Simplify the right side: \( 36 \)
So we have \( 2x = 36 \)

Step 2: Solve for \( x \)

Divide both sides of \( 2x = 36 \) by 2.
\( x = \frac{36}{2} = 18 \)

Step 3: Substitute \( x = 18 \) into \( x + y = 30 \)

Substitute \( x = 18 \) into \( x + y = 30 \): \( 18 + y = 30 \)

Step 4: Solve for \( y \)

Subtract 18 from both sides: \( y = 30 - 18 = 12 \)

Step 1: Add the two equations

Add the equations \( 3x + 6y = 6 \) and \( 4x - 6y = 8 \) to eliminate \( y \).
\( (3x + 6y) + (4x - 6y) = 6 + 8 \)
Simplify the left side: \( 3x + 6y + 4x - 6y = 7x \)
Simplify the right side: \( 14 \)
So we have \( 7x = 14 \)

Step 2: Solve for \( x \)

Divide both sides of \( 7x = 14 \) by 7.
\( x = \frac{14}{7} = 2 \)

Step 3: Substitute \( x = 2 \) into \( 3x + 6y = 6 \)

Substitute \( x = 2 \) into \( 3x + 6y = 6 \): \( 3(2) + 6y = 6 \)
Simplify: \( 6 + 6y = 6 \)

Step 4: Solve for \( y \)

Subtract 6 from both sides: \( 6y = 6 - 6 = 0 \)
Divide both sides by 6: \( y = \frac{0}{6} = 0 \)

Answer:

\( x = 18 \), \( y = 12 \)

Problem 4: Solve the system \(

$$\begin{cases} 3x + 6y = 6 \\ 4x - 6y = 8 \end{cases}$$

\)