Question

problem 5
these two lines are parallel. write an equation for each.

Explanation:

Step1: Determine the slope of the lines

Since the lines are parallel, they have the same slope. Let's take the first line (passing through the origin \((0,0)\) and, for example, \((2,2)\)). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{2 - 0}=1\). Wait, maybe I made a mistake. Let's check the second line. The second line passes through \((0, - 3.2)\) (or \((0,-\frac{16}{5})\)) and, for example, \((4,0)\). The slope \(m=\frac{0-(-\frac{16}{5})}{4 - 0}=\frac{\frac{16}{5}}{4}=\frac{4}{5} = 0.8\)? Wait, maybe the grid is such that each square is 1 unit. Let's re - examine. The first line passes through \((0,0)\) and \((5,5)\)? No, looking at the graph, the first line (the upper one) passes through \((0,0)\) and when \(x = 5\), \(y = 5\)? Wait, no, maybe the slope is \(\frac{1}{1}\)? Wait, no, let's take two points on the first line. Let's take \((0,0)\) and \((1,1)\), so slope \(m = 1\)? Wait, the second line: when \(x = 4\), \(y=0\) and when \(x = 0\), \(y=-3.2\). Wait, maybe the slope is \(\frac{0 - (-3.2)}{4-0}=\frac{3.2}{4}=0.8=\frac{4}{5}\). Wait, maybe I misread the points. Let's start over.

The equation of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

For the first line (the one passing through the origin \((0,0)\)):
The y - intercept \(b = 0\). To find the slope, we can take two points \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(5,5)\)? Wait, no, looking at the graph, when \(x = 1\), \(y = 1\), so slope \(m=\frac{1 - 0}{1 - 0}=1\)? Wait, no, maybe the slope is \(\frac{4}{5}\). Wait, let's check the second line. The second line passes through \((0,-3.2)\) and \((4,0)\). The slope \(m=\frac{0-(-3.2)}{4 - 0}=\frac{3.2}{4}=0.8=\frac{4}{5}\). Now, let's check the first line. If the first line passes through \((0,0)\) and \((5,4)\), then slope \(m=\frac{4 - 0}{5 - 0}=\frac{4}{5}\). Ah, that makes sense because parallel lines have the same slope. So the slope \(m=\frac{4}{5}=0.8\).

Step2: Find the equation of the first line (upper line)

The first line passes through the origin \((0,0)\), so the y - intercept \(b = 0\). Using the slope - intercept form \(y=mx + b\), with \(m = \frac{4}{5}\) and \(b = 0\), the equation is \(y=\frac{4}{5}x\) or \(y = 0.8x\).

For the second line (the lower line):
The y - intercept \(b=-3.2\) (or \(b =-\frac{16}{5}\)). Since the lines are parallel, the slope \(m=\frac{4}{5}\) (same as the first line). Using the slope - intercept form \(y=mx + b\), we substitute \(m=\frac{4}{5}\) and \(b=-3.2\) (or \(b =-\frac{16}{5}\)). So the equation is \(y=\frac{4}{5}x-3.2\) (or \(y = 0.8x-3.2\)).

Wait, maybe I made a mistake in the slope calculation. Let's take two points on the first line: \((0,0)\) and \((5,4)\) (since when \(x = 5\), \(y = 4\) to be parallel to the second line which goes from \((0,-3.2)\) to \((4,0)\) (because \(\frac{0 - (-3.2)}{4-0}=\frac{3.2}{4}=0.8=\frac{4}{5}\)) and \(\frac{4 - 0}{5 - 0}=\frac{4}{5}\)). So the slope \(m=\frac{4}{5}\).

So for the first line (passing through \((0,0)\)): \(y=\frac{4}{5}x\) (or \(y = 0.8x\)).

For the second line (passing through \((0,-3.2)\)): \(y=\frac{4}{5}x-3.2\) (or \(y = 0.8x-3.2\)).

Answer:

The equations of the two parallel lines are \(y = 0.8x\) (or \(y=\frac{4}{5}x\)) and \(y=0.8x - 3.2\) (or \(y=\frac{4}{5}x-\frac{16}{5}\)).