Question

y - 4 = -(x - 2)
graph of line b:
graph of a line on a coordinate plane
answer attempt 1 out of 2
the slope of line a is blank and the slope of line b is blank. therefore the slope of

Explanation:

Step1: Find slope of Line A

The equation of Line A is \( y - 4 = -(x - 2) \). The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope. Comparing \( y - 4 = - 1\times(x - 2) \) with the point - slope form, we can see that the slope of Line A, \( m_A=-1 \).

Step2: Find slope of Line B

To find the slope of Line B, we can use two points on the line. From the graph, we can see that the line passes through the points \( (0,4) \) (the y - intercept) and \( (8,0) \) (the x - intercept). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(0,4) \) and \( (x_2,y_2)=(8,0) \). Then \( m_B=\frac{0 - 4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \)? Wait, no, wait. Wait, let's check another pair of points. Wait, when \( x = 0 \), \( y = 4 \); when \( x = 2 \), \( y=3 \); when \( x = 4 \), \( y = 2 \); when \( x=8 \), \( y = 0 \). Let's use \( (0,4) \) and \( (2,3) \). Then \( m=\frac{3 - 4}{2 - 0}=\frac{-1}{2} \)? Wait, no, that's a mistake. Wait, the line is decreasing. Wait, let's take two points: \( (0,4) \) and \( (8,0) \). The change in \( y \) is \( 0 - 4=-4 \), the change in \( x \) is \( 8 - 0 = 8 \). So the slope \( m=\frac{-4}{8}=-\frac{1}{2} \)? Wait, no, wait the equation of Line A is \( y-4=-(x - 2) \), which simplifies to \( y=-x + 2+4=-x + 6 \)? Wait, no, \( y-4=-x + 2\), so \( y=-x+6 \). Wait, the graph of Line B: when \( x = 0 \), \( y = 4 \); when \( x = 2 \), \( y=3 \); when \( x = 4 \), \( y = 2 \); when \( x=6 \), \( y = 1 \); when \( x = 8 \), \( y=0 \). So the slope between \( (0,4) \) and \( (8,0) \) is \( \frac{0 - 4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \)? Wait, but the equation of Line A is \( y=-x + 6 \) (since \( y-4=-(x - 2)\Rightarrow y=-x + 2 + 4=-x+6 \)). Wait, maybe I made a mistake in the points. Wait, let's re - examine the graph. The line in the graph: when \( x=- 8 \), \( y = 8 \); when \( x = 0 \), \( y = 4 \); when \( x = 8 \), \( y = 0 \). So the slope between \( (-8,8) \) and \( (0,4) \): \( m=\frac{4 - 8}{0-(-8)}=\frac{-4}{8}=-\frac{1}{2} \). Wait, but the equation of Line A is \( y=-x + 6 \), so its slope is - 1. For Line B, let's take two points: \( (0,4) \) and \( (2,3) \): \( m=\frac{3 - 4}{2-0}=\frac{-1}{2} \), \( (2,3) \) and \( (4,2) \): \( m=\frac{2 - 3}{4 - 2}=\frac{-1}{2} \), \( (4,2) \) and \( (6,1) \): \( m=\frac{1 - 2}{6 - 4}=\frac{-1}{2} \), \( (6,1) \) and \( (8,0) \): \( m=\frac{0 - 1}{8 - 6}=\frac{-1}{2} \). So the slope of Line B is \( -\frac{1}{2} \)? Wait, but maybe the question is to find the slope of Line A from its equation and slope of Line B from the graph. Wait, the equation of Line A: \( y-4=-(x - 2) \), so in point - slope form, slope is - 1. For Line B, using two points \( (0,4) \) and \( (8,0) \), slope \( m=\frac{0 - 4}{8 - 0}=- \frac{4}{8}=-\frac{1}{2} \)? Wait, no, that can't be. Wait, maybe I misread the graph. Wait, the line in the graph: when \( x = 0 \), \( y = 4 \); when \( x = 2 \), \( y = 3 \); so the run is 2, rise is - 1, so slope is \( \frac{-1}{2} \). But the equation of Line A is \( y=-x + 6 \), so slope - 1.

Wait, maybe the problem is that the user's question is about the slopes of Line A (from the equation \( y - 4=-(x - 2) \)) and Line B (from the graph). Let's re - do the slope of Line A:

The point - slope form is \( y - y_1=m(x - x_1) \), where \( m \) is the slope. So for \( y - 4=-(x - 2) \), the slope \( m_A=-1 \).

For Line B: Let's take two points on the line. From the graph, when \( x = 0 \), \( y = 4 \); when \( x = 8 \), \( y = 0 \). So using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), with \( (x_1,y_1)=(0,4) \) and \( (x_2,y_2)=(8,0) \), we have \( m_B=\frac{0 - 4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \)? Wait, no, that's incorrect. Wait, if we take \( (0,4) \) and \( (2,3) \), \( m=\frac{3 - 4}{2 - 0}=\frac{-1}{2} \), \( (2,3) \) and \( (4,2) \), \( m=\frac{2 - 3}{4 - 2}=\frac{-1}{2} \), so the slope is \( -\frac{1}{2} \). But the equation of Line A is \( y=-x + 6 \), slope - 1.

Wait, maybe I made a mistake in the graph. Wait, the line in the graph: when \( x=-8 \), \( y = 8 \); \( x=-6 \), \( y = 7 \); \( x=-4 \), \( y = 6 \); \( x=-2 \), \( y = 5 \); \( x = 0 \), \( y = 4 \); \( x = 2 \), \( y = 3 \); \( x = 4 \), \( y = 2 \); \( x = 6 \), \( y = 1 \); \( x = 8 \), \( y = 0 \). So the change in \( y \) when \( x \) increases by 2 is - 1. So the slope is \( \frac{-1}{2} \). But the equation of Line A is \( y-4=-(x - 2) \), which is \( y=-x + 6 \), slope - 1.

Wait, but maybe the question is to find the slope of Line A from its equation (which is - 1) and the slope of Line B from the graph. Let's confirm the slope of Line B again. Let's use two points: \( (0,4) \) and \( (8,0) \). The slope \( m=\frac{0 - 4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \). Wait, but that seems different from Line A's slope. But maybe I misread the equation. Wait, the equation of Line A: \( y - 4=-(x - 2) \), expand it: \( y-4=-x + 2 \), so \( y=-x+6 \). So slope is - 1. For Line B, using \( (0,4) \) and \( (2,3) \), slope is \( \frac{3 - 4}{2 - 0}=-\frac{1}{2} \).

Answer:

The slope of Line A is \(-1\) and the slope of Line B is \(-\frac{1}{2}\)