Question

which statement about the following system is correct?
y = 2x - 5
y = 2x + 5

• the equations are independent because the lines intersect in one point.
• the equations are independent because the lines are parallel.
• the equations are dependent because the lines do not intersect.
• the equations are dependent because the lines are the same line.

Explanation:

Step1: Analyze the slopes of the lines

The two equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the equation \(y=2x - 5\), the slope \(m_1 = 2\) and the y - intercept \(b_1=- 5\). For the equation \(y = 2x+5\), the slope \(m_2 = 2\) and the y - intercept \(b_2 = 5\). Since \(m_1=m_2\) (both slopes are 2) and \(b_1 eq b_2\) ( - 5≠5), the two lines are parallel. Parallel lines do not intersect.

Step2: Recall the definitions of independent and dependent equations

- Independent equations: Two equations are independent if their graphs (lines) intersect at exactly one point.
- Dependent equations: Two equations are dependent if their graphs (lines) are the same (coincident) line.

Since the two lines are parallel (do not intersect) and not the same line, we analyze the options:
- Option 1: Says lines intersect at one point, but parallel lines don't intersect. So this is wrong.
- Option 2: The equations are independent because the lines are parallel. Wait, no. Wait, independent equations have a unique solution (intersect at one point), dependent have infinitely many (same line). But parallel lines (non - coincident) have no solution. But in terms of linear equations, two linear equations \(a_1x + b_1y=c_1\) and \(a_2x + b_2y = c_2\) are independent if the system has a unique solution, dependent if it has infinitely many solutions. For two lines \(y = m_1x + b_1\) and \(y=m_2x + b_2\), if \(m_1 eq m_2\), independent (one solution). If \(m_1 = m_2\) and \(b_1=b_2\), dependent (infinitely many solutions). If \(m_1=m_2\) and \(b_1 eq b_2\), the system has no solution. But the question's options: Wait, maybe the question's use of independent and dependent is a bit different. Wait, the second option says "The equations are independent because the lines are parallel." Wait, no, let's re - check the options. Wait, maybe there is a mistake in my initial thought. Wait, no, let's re - evaluate.

Wait, the two equations are distinct (different y - intercepts) and have the same slope. So they are parallel. Now, the options:

Wait, the second option: "The equations are independent because the lines are parallel." Wait, no, independent equations should have a solution (intersect). But parallel lines (non - coincident) have no solution. But maybe the question's definition of independent is that the equations are not equivalent (not the same line). Dependent equations are equivalent (same line). So if two equations are not the same line (even if parallel), they are independent? Wait, that might be the case here. Let's check the options again:

- Option 1: Lines intersect at one point: wrong, parallel lines don't intersect.
- Option 2: Equations are independent because lines are parallel. Since the lines are not the same (so the equations are not equivalent), they are independent. And parallel lines (non - coincident) are not the same line, so the equations are independent.
- Option 3: Equations are dependent because lines do not intersect: Dependent equations have infinitely many solutions (same line), but these lines are not the same. So wrong.
- Option 4: Equations are dependent because lines are the same: Lines are not the same (different y - intercepts). So wrong.

So the correct option is the second one: The equations are independent because the lines are parallel. Wait, but according to the standard definition of independent and dependent in linear systems:

A system of linear equations is:

- Consistent and independent: One solution (intersecting lines, \(m_1 eq m_2\))
- Consistent and dependent: Infinitely many solutions (same line, \(m_1 = m_2\), \(b_1=b_2\))
- Inconsistent: No solution (parallel lines, \(m_1 = m_2\), \(b_1 eq b_2\))

But the question's options use "independent" and "dependent" in a way that if the equations are not the same (even if parallel), they are independent. So among the options, the second option is the only one that can be considered correct. Because the other options are clearly wrong.

Answer:

The equations are independent because the lines are parallel. (The option: "The equations are independent because the lines are parallel.")