Question

graph of a line with two points marked. the line passes through (0, -2) and (2, 5)? wait, no, looking at the grid: one point is at (0, -2)? wait, no, the y-axis has 0, 5, -5. wait, the lower point is at (0, -2)? wait, no, the grid: x from -5 to 5, y from -5 to 5. the line has a point at (0, -2)? wait, no, the black dot at (0, -2)? wait, no, the image shows a line with a dot at (0, -2) and another at (2, 5)? wait, no, the upper dot is at (2, 5)? wait, the options are: y = (2/7)x + 2, y = (7/2)x + 2, y = (2/7)x - 2, y = (7/2)x - 2. wait, maybe the points are (0, -2) and (2, 5)? wait, no, lets re-express: the line has a y-intercept? wait, the lower dot is at (0, -2), and the upper dot is at (2, 5)? wait, no, the x-coordinate of the upper dot: from 0, moving 2 units right (since grid is 1 unit per square), so x=2, y=5? wait, no, 0 to 5 on y is 5 units, so each grid square is 1 unit. so the two points: lets see, the line passes through (0, -2) and (2, 5)? wait, no, the slope would be (5 - (-2))/(2 - 0) = 7/2. so the equation would be y = (7/2)x - 2? wait, no, if it passes through (0, -2), then y-intercept is -2, so y = mx - 2. then slope m: from (0, -2) to (2, 5), rise is 7, run is 2, so m=7/2. so the equation is y = (7/2)x - 2. but the options are: 1. y = (2/7)x + 2, 2. y = (7/2)x + 2, 3. y = (2/7)x - 2, 4. y = (7/2)x - 2. wait, maybe the lower dot is (0, -2) and the upper dot is (2, 5)? wait, no, maybe (0, -2) and (2, 5) is wrong. wait, the image: the vertical line is y-axis, horizontal is x-axis. the lower dot is at (0, -2) (since its on y-axis, x=0, y=-2), and the upper dot is at (2, 5)? wait, no, 0 to 5 on y is 5, so y=5 when x=2? then slope is (5 - (-2))/(2 - 0) = 7/2. so the equation is y = (7/2)x - 2. so the options include that. so the problem is to find the equation of the line given its graph (with two points) and multiple-choice options. so the ocr text is: the graph of a line with two points (probably (0, -2) and (2, 5)) and four options: y = (2/7)x + 2, y = (7/2)x + 2, y = (2/7)x - 2, y = (7/2)x - 2. (note: the original images text is as follows, with the line equation options: first option ( y = \frac{2}{7}x + 2 ), second ( y = \frac{7}{2}x + 2 ), third ( y = \frac{2}{7}x - 2 ), fourth ( y = \frac{7}{2}x - 2 ). the graph has a coordinate plane with x from -5 to 5, y from -5 to 5, a line with a dot at (0, -2) and another at (2, 5) (assuming grid squares are 1 unit each).)

Explanation:

Step1: Identify two points on the line

From the graph, we can see two points: when \( x = 0 \), \( y=-2 \) (the y - intercept), and another point \( (2,5) \)? Wait, no, let's check again. Wait, the lower point is at \( (0, - 2) \)? Wait, no, looking at the grid, the vertical line is the y - axis. The lower black dot is at \( (0,-2) \)? Wait, no, the grid lines: the y - axis has 0, then down to - 5. Wait, maybe the two points are \( (0, - 2) \) and \( (2,5) \)? Wait, no, let's calculate the slope. Wait, the upper dot: let's see the x - coordinate. From the origin (0,0), moving 2 units to the right (x = 2) and y = 5? Wait, no, maybe the two points are \( (0,-2) \) and \( (2,5) \)? Wait, no, let's do it properly. The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the two points: one is \( (0,-2) \) (the y - intercept) and another point, say, when x = 2, y = 5? Wait, no, looking at the line, when x = 2, y = 5? Wait, the upper dot: x is 2 (since from 0, moving 2 units right) and y is 5 (moving 5 units up). So the two points are \( (0,-2) \) and \( (2,5) \)? Wait, no, wait the lower dot is at (0, - 2)? Wait, the y - axis: the line crosses the y - axis at (0, - 2)? Wait, no, the grid: the horizontal line is y = 0 (the x - axis). The lower black dot is below the x - axis, at (0, - 2)? Wait, maybe I made a mistake. Wait, let's check the options. The options have slopes \( \frac{2}{7} \), \( \frac{7}{2} \), etc. Let's recast. Let's take two points: (0, - 2) and (2,5)? No, wait, the upper dot: x = 2, y = 5? Wait, the vertical distance between (0, - 2) and (2,5) is \( 5-(-2)=7 \), and horizontal distance is \( 2 - 0 = 2 \), so slope \( m=\frac{7}{2} \). And the y - intercept \( b=-2 \)? Wait, no, the y - intercept is the value of y when x = 0. If the line crosses the y - axis at (0, - 2), then \( b=-2 \). So the equation of the line is \( y=mx + b \), where \( m=\frac{7}{2} \) and \( b=-2 \), so \( y=\frac{7}{2}x-2 \). Wait, let's check the options. The last option is \( y = \frac{7}{2}x-2 \). Let's verify with the points. For x = 0, \( y=\frac{7}{2}(0)-2=-2 \), which matches the lower point. For x = 2, \( y=\frac{7}{2}(2)-2=7 - 2 = 5 \), which matches the upper point (2,5). So that's correct.

Step2: Match with the options

We found that the slope \( m = \frac{7}{2} \) and the y - intercept \( b=-2 \), so the equation is \( y=\frac{7}{2}x - 2 \), which is the last option.

Answer:

\( y=\frac{7}{2}x - 2 \) (the last option, assuming the options are ordered as given: the fourth option is \( y=\frac{7}{2}x-2 \))