Question

1. the graph of a line is shown on the coordinate grid.

which equation represents the line?
( y = 2x - 5 )
( 8.5x - y = -5 )
( y - 2 = \frac{3}{5}(x - 12) )
( y - 4 = \frac{5}{3}(x - 2) )

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -5)\) (the y - intercept) and we can find another point. Let's take \((10, 1)\) (by looking at the grid, when \(x = 10\), \(y=1\)).

Step2: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((x_1,y_1)=(0, - 5)\) and \((x_2,y_2)=(10,1)\), we have \(m=\frac{1-(-5)}{10 - 0}=\frac{6}{10}=\frac{3}{5}\).

Step3: Check each option

• Option 1: \(y = 2x-5\). The slope here is \(2\), which is not equal to \(\frac{3}{5}\), so this is incorrect.
• Option 2: Rewrite \(8.5x - y=-5\) in slope - intercept form (\(y=mx + b\)). We get \(y = 8.5x + 5\). The slope is \(8.5=\frac{17}{2}\), not \(\frac{3}{5}\), so this is incorrect.
• Option 3: The equation \(y - 2=\frac{3}{5}(x - 12)\) is in point - slope form \(y - y_1=m(x - x_1)\), where \(m = \frac{3}{5}\) and the point is \((12,2)\). Let's check if \((12,2)\) is on the line. Using the slope \(\frac{3}{5}\) and the y - intercept \((0,-5)\), when \(x = 12\), \(y=\frac{3}{5}\times12-5=\frac{36}{5}-5=\frac{36 - 25}{5}=\frac{11}{5}=2.2\)? Wait, no, let's substitute \(x = 12\) into the equation \(y-2=\frac{3}{5}(12 - 12)\), then \(y-2 = 0\), so \(y = 2\). And let's check the slope between \((0,-5)\) and \((12,2)\): \(m=\frac{2-(-5)}{12-0}=\frac{7}{12}\)? No, wait, earlier we found the slope between \((0,-5)\) and \((10,1)\) is \(\frac{3}{5}\). Let's check the slope between \((12,2)\) and \((0,-5)\): \(\frac{2-(-5)}{12 - 0}=\frac{7}{12}\)? Wait, maybe I made a mistake in the point. Wait, let's use the point - slope form. The slope of the line is \(\frac{3}{5}\), and if we take the point \((12,2)\), let's see if the line passes through it. Using the slope - intercept form \(y=\frac{3}{5}x-5\), when \(x = 12\), \(y=\frac{3}{5}\times12-5=\frac{36}{5}-5=\frac{36 - 25}{5}=\frac{11}{5}=2.2\), which is close to \(2\) (maybe due to grid approximation). Wait, let's check the third option's slope. The equation \(y - 2=\frac{3}{5}(x - 12)\) has a slope of \(\frac{3}{5}\), which matches our calculated slope.
• Option 4: The equation \(y - 4=\frac{5}{3}(x - 2)\) has a slope of \(\frac{5}{3}\), which is not equal to \(\frac{3}{5}\), so this is incorrect.

Answer:

\(y - 2=\frac{3}{5}(x - 12)\) (the third option)