Question

use the graph to determine the number of solutions the system has. then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. y = 0.5x y = x + 2 graph with equations 2x + 3y = 10, 2x + 3y = 6, y = 0.5x, y = x + 2, 4x + 6y = 12, 2x - 2y = -4, -10x - 5y = 15 a) 1 solution; consistent and independent b) infinitely many solutions; consistent and dependent c) 1 solution; consistent and dependent d) no solution; inconsistent

Explanation:

Step1: Analyze the equations' slopes

The equation \( y = 0.5x \) has a slope of \( 0.5 \), and \( y = x + 2 \) has a slope of \( 1 \). Since the slopes are different, the lines are not parallel and will intersect at one point.

Step2: Determine solution count

Two lines with different slopes intersect at exactly one point, so the system has 1 solution.

Step3: Classify consistency and dependence

A system with at least one solution is consistent. Since the lines are not coinciding (different slopes, different equations), they are independent. Wait, but wait—wait, no, wait: Wait, the original selected option was C, but let's re - check. Wait, no, the correct analysis: For a system of linear equations \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \), if \( m_1
eq m_2 \), the system has one solution, is consistent (because there is a solution), and independent (because the equations are not equivalent, i.e., not the same line). So the correct option should be A. Wait, the initial selected option in the image was C, but that's wrong. Let's correct:

The two lines \( y = 0.5x \) (slope \( 0.5 \), y - intercept \( 0 \)) and \( y=x + 2 \) (slope \( 1 \), y - intercept \( 2 \)) have different slopes and different y - intercepts. So they intersect at one point (1 solution). A consistent system has at least one solution, and independent systems have different equations (not dependent, where dependent would be the same line, i.e., same slope and same y - intercept). So the system is consistent (because there is a solution) and independent (because the two equations represent different lines). So the correct option is A.

Answer:

A) 1 solution; consistent and independent