Question

question 6
which of the following systems has equations that are dependent?
○ 3x - 3y = 9 ○ y = -3x - 1 ○ 5x - y = 10 ○ 2x - y = 5
3x - 3y = 6 y = -3x - 2 5x + y = 10 4x - 2y = 10

Explanation:

To determine if a system of linear equations is dependent, the two equations must be scalar multiples of each other (i.e., they represent the same line, so there are infinitely many solutions). Let's analyze each option:

Option 1: \( 3x - 3y = 9 \) and \( 3x - 3y = 6 \)

• Simplify the first equation: Divide by 3: \( x - y = 3 \) or \( y = x - 3 \)
• Simplify the second equation: Divide by 3: \( x - y = 2 \) or \( y = x - 2 \)
• These are parallel lines (same slope, different y-intercepts), so they are inconsistent (no solution), not dependent.

Option 2: \( y = -3x - 1 \) and \( y = -3x - 2 \)

• Both are in slope-intercept form (\( y = mx + b \)). The slopes (\( m = -3 \)) are the same, but the y-intercepts (\( b = -1 \) and \( b = -2 \)) are different. These are parallel lines (no solution), not dependent.

Option 3: \( 5x - y = 10 \) and \( 5x + y = 10 \)

• Rewrite the first equation: \( y = 5x - 10 \)
• Rewrite the second equation: \( y = -5x + 10 \)
• The slopes are \( 5 \) and \( -5 \) (different), so they intersect at one point (unique solution), not dependent.

Option 4: \( 2x - y = 5 \) and \( 4x - 2y = 10 \)

• Simplify the first equation: Solve for \( y \): \( y = 2x - 5 \)
• Simplify the second equation: Divide by 2: \( 2x - y = 5 \), which is the same as the first equation. So \( 4x - 2y = 10 \) is a scalar multiple (multiplied by 2) of \( 2x - y = 5 \). They represent the same line, so the system is dependent.

Answer:

The system with equations \( 2x - y = 5 \) and \( 4x - 2y = 10 \) (the last option) has dependent equations.