Question

write the equation of the line in fully simplified slope - intercept form.

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, 12)\) (wait, no, let's check again. Wait, the y-intercept? Wait, actually, looking at the graph, when \(x = 0\), what's \(y\)? Wait, no, let's find two clear points. Let's take \((2, 10)\) and \((6, 6)\)? Wait, no, maybe better to take \((0, 12)\) and \((6, 6)\)? Wait, no, let's check the grid. Wait, the line crosses the y-axis at \((0, 12)\)? Wait, no, the y-axis is the vertical axis. Wait, the x-axis is horizontal, y-axis vertical. Wait, the points: let's see, when \(x = 2\), \(y = 10\)? Wait, no, maybe I got x and y mixed. Wait, the horizontal axis is y? Wait, no, the labels: the horizontal axis is labeled y (from -12 to 12), vertical axis labeled x (from -12 to 12). So it's a rotated coordinate system? Wait, no, maybe the graph has x and y swapped. Wait, usually, x is horizontal, y vertical, but here the horizontal axis is y, vertical is x. So the coordinates: (x, y) where x is vertical, y is horizontal. So let's find two points. Let's take when x = 2 (vertical), y = 10 (horizontal)? Wait, no, the blue line: let's find two points. Let's see, when x = 2, y = 10? Wait, no, maybe the points are (2, 10) and (6, 6)? Wait, no, let's check the slope. Wait, the line goes from (2, 10) to (6, 6)? Wait, no, let's look at the grid. Each square is 1 unit. Let's take two points: (2, 10) and (6, 6). Wait, the change in x (vertical) is 6 - 2 = 4, change in y (horizontal) is 6 - 10 = -4. So slope (m) is (change in x)/(change in y) if x is vertical, but usually slope is (change in y)/(change in x). Wait, maybe I have x and y swapped. Let's assume that the horizontal axis is x, vertical is y (maybe the labels are reversed). Let's reorient: horizontal axis is x, vertical is y. Then the points: when x = 2, y = 10? No, the blue line: let's see, the line crosses the y-axis (vertical) at (0, 12)? Wait, no, the vertical axis is x, horizontal is y. So (x, y) where x is vertical, y is horizontal. So the line passes through (2, 10) and (6, 6). So x1 = 2, y1 = 10; x2 = 6, y2 = 6. Then the slope (m) is (y2 - y1)/(x2 - x1) = (6 - 10)/(6 - 2) = (-4)/4 = -1. Wait, no, slope is (change in y)/(change in x), but if x is vertical, then change in x is vertical, change in y is horizontal. So maybe the slope is (change in x)/(change in y) because x is vertical. Wait, this is confusing. Alternatively, maybe the graph has x and y axes swapped. Let's assume that the horizontal axis is x, vertical is y (standard), and the labels are misprinted. Then the line passes through (0, 12) and (6, 6). Wait, when x = 0, y = 12; when x = 6, y = 6. Then the slope (m) is (6 - 12)/(6 - 0) = (-6)/6 = -1. Then the y-intercept (b) is 12 (when x = 0, y = 12). So the equation is y = mx + b = -x + 12? Wait, no, when x = 6, y = 6: 6 = -6 + 12? 6 = 6, yes. When x = 2, y = 10: 10 = -2 + 12? 10 = 10, yes. So that works. So the equation is y = -x + 12? Wait, no, wait, if x is horizontal, y vertical, then the line has slope -1, y-intercept 12. Wait, but in the graph, the horizontal axis is labeled y, vertical x. So maybe the equation is x = -y + 12, but we need slope-intercept form, which is y = mx + b. So if we consider y as the dependent variable (horizontal), x as independent (vertical), then we need to solve for y. Wait, slope-intercept form is y = mx + b, where m is slope, b is y-intercept (when x = 0, y = b). So let's take x as vertical (independent), y as horizontal (dependent). So two points: (x1, y1) = (2, 10), (x2, y2) = (6, 6). Then slope m = (y2 - y1)/(x2 - x1) = (6 - 10)/(6 - 2) = (-4)/4 = -1. Then using point-slope form: y - y1 = m(x - x1). Using (2, 10): y - 10 = -1(x - 2) => y - 10 = -x + 2 => y = -x + 12. Let's check with (6, 6): y = -6 + 12 = 6, which matches. So the equation is y = -x + 12. Wait, but let's confirm with another point. When x = 0, y = 12: yes, that's on the line. When x = 10, y = 2: y = -10 + 12 = 2, which is on the line (x = 10, y = 2). So that works.

Step2: Write the slope-intercept form

Slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept (when \(x = 0\), \(y = b\)). We found \(m = -1\) and \(b = 12\). So the equation is \(y = -x + 12\). Wait, but let's check the graph again. If the horizontal axis is y, vertical is x, then when x = 0 (vertical axis), y = 12 (horizontal axis), which is the y-intercept. The slope is -1, so for each unit increase in x (vertical), y (horizontal) decreases by 1. Which matches the points. So the equation is \(y = -x + 12\). Wait, but maybe I made a mistake in x and y. Wait, no, the problem says "slope-intercept form", which is \(y = mx + b\), where \(y\) is the dependent variable (horizontal axis here), \(x\) is the independent variable (vertical axis). So yes, \(y = -x + 12\).

Answer:

\(y = -x + 12\)