Question

1. leveled practice two rovers are exploring a planet. the system of equations below shows each rover’s elevation, y, at time x. what conclusion can you reach about the system of equations? graph of two lines, rover b (blue) and rover a (red), on a coordinate plane with x from -8 to 8 and y from -4 to 4 the slope for the rover a equation is blank the slope for the rover b equation. the y-intercepts of the equations are blank. the system of equations has blank solution(s).

Explanation:

Step1: Analyze Slopes

For a line, slope \( m = \frac{\Delta y}{\Delta x} \).

• Rover B (blue line): Passes through \((0,0)\) and \((1,2)\) (approx, since it's steeper). Slope \( m_B = \frac{2 - 0}{1 - 0} = 2 \).
• Rover A (red line): Passes through \((4,0)\) and \((5,1)\) (approx, less steep). Slope \( m_A = \frac{1 - 0}{5 - 4} = 1 \).

So \( m_A < m_B \) (or "less than").

Step2: Analyze y - intercepts

y - intercept is where \( x = 0 \).

• Rover B: Crosses y - axis at \((0,0)\), so y - intercept \( = 0 \).
• Rover A: Crosses y - axis at \((0, - 4)\) (from graph, when \( x = 0 \), \( y=-4\))? Wait, no—wait, Rover A's line: when \( x = 4 \), \( y = 0 \); when \( x = 0 \), let's see. Wait, maybe I misread. Wait, Rover B passes through \((0,0)\) (origin), Rover A passes through \((4,0)\) and when \( x = 0 \), what's \( y \)? Let's calculate slope of Rover A: from \((4,0)\) to \((8,4)\) (since it goes up 4 when x increases by 4), so slope \( m_A=\frac{4 - 0}{8 - 4}=1\). Rover B: from \((0,0)\) to \((2,4)\), slope \( m_B=\frac{4 - 0}{2 - 0}=2\). So y - intercept of Rover B is \( 0 \) (at \( x = 0 \), \( y = 0 \)), y - intercept of Rover A: when \( x = 0 \), using \( y - y_1=m(x - x_1) \) for Rover A: \( y - 0 = 1(x - 4) \), so \( y=x - 4 \). So at \( x = 0 \), \( y=-4 \). So y - intercepts are different (Rover B: 0, Rover A: - 4).

Step3: Analyze solutions to system

Parallel lines? No, slopes are different (1 vs 2), so they intersect at one point. So system has one solution.

Answer:

The slope for the Rover A equation is \(\boldsymbol{\text{less than}}\) the slope for the Rover B equation.
The y - intercepts of the equations are \(\boldsymbol{\text{different}}\).
The system of equations has \(\boldsymbol{\text{one}}\) solution(s).

(Note: For the first blank, "less than" (or "smaller than"); second blank "different"; third blank "one".)