Question

the graph of an equation is sketched in the figure. describe five ordered - pair solutions of this equation by using a table. (simplify your answers.) \

$$\begin{tabular}{|c|c|} \\hline x & y \\\\ \\hline -6 & \\square \\\\ \\hline -3 & \\square \\\\ \\hline 0 & \\square \\\\ \\hline 3 & \\square \\\\ \\hline 6 & \\square \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Find the equation of the line

First, identify two points on the line. From the graph, we can see that the line passes through \((0, 2)\) and \((8, 0)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 2}{8 - 0}=-\frac{1}{4}\). Using the slope - intercept form \(y = mx + b\) (where \(b\) is the \(y\) - intercept), since \(b = 2\) (the line crosses the \(y\) - axis at \((0,2)\)), the equation of the line is \(y=-\frac{1}{4}x + 2\).

Step2: Calculate \(y\) for \(x=-6\)

Substitute \(x = - 6\) into the equation \(y=-\frac{1}{4}x + 2\).
\(y=-\frac{1}{4}\times(-6)+2=\frac{6}{4}+2=\frac{3}{2}+2=\frac{3 + 4}{2}=\frac{7}{2}=3.5\)

Step3: Calculate \(y\) for \(x = - 3\)

Substitute \(x=-3\) into the equation \(y =-\frac{1}{4}x+2\).
\(y=-\frac{1}{4}\times(-3)+2=\frac{3}{4}+2=\frac{3 + 8}{4}=\frac{11}{4}=2.75\) Wait, no, let's re - check. Wait, maybe we made a mistake in the slope. Let's re - calculate the slope. Looking at the graph, when \(x = 0\), \(y = 2\); when \(x = 8\), \(y=0\). Also, when \(x=-8\), let's see the \(y\) - value. Wait, maybe a better way is to look at the grid. Each square is 1 unit. Let's take two points: \((- 8,4)\) and \((0,2)\) and \((8,0)\). The slope between \((0,2)\) and \((8,0)\) is \(\frac{0 - 2}{8 - 0}=-\frac{1}{4}\), correct. Wait, when \(x=-6\), let's count the grid. From \(x = 0\) ( \(y = 2\) ), moving 6 units to the left ( \(x=-6\) ), since the slope is \(-\frac{1}{4}\), for every 4 units we move to the right, \(y\) decreases by 1. So for 6 units to the left (which is equivalent to - 6 units in the \(x\) - direction), the change in \(y\) is \(\frac{6}{4}=\frac{3}{2}\). So \(y=2+\frac{3}{2}=\frac{4 + 3}{2}=\frac{7}{2}=3.5\) or \(\frac{7}{2}\).

Wait, maybe an easier way is to use the graph. Let's look at the \(x\) - value of \(-6\). On the graph, when \(x=-6\), we can see that the \(y\) - value is 3.5 (or \(\frac{7}{2}\)).

Wait, let's re - do the equation. Let's take two points: \((- 8,4)\) and \((0,2)\). The slope \(m=\frac{2 - 4}{0-(-8)}=\frac{-2}{8}=-\frac{1}{4}\), correct. So the equation is \(y =-\frac{1}{4}x + 2\).

For \(x=-6\): \(y=-\frac{1}{4}\times(-6)+2=\frac{6}{4}+2 = 1.5+2=3.5\)

For \(x=-3\): \(y=-\frac{1}{4}\times(-3)+2=\frac{3}{4}+2 = 0.75 + 2=2.75\)? Wait, no, that can't be. Wait, maybe the line passes through \((- 4,3)\), \((0,2)\), \((4,1)\), \((8,0)\). Ah! I see my mistake. The slope is \(\frac{2 - 3}{0-(-4)}=\frac{-1}{4}=-\frac{1}{4}\), but when \(x=-4\), \(y = 3\); \(x = 0\), \(y = 2\); \(x = 4\), \(y = 1\); \(x = 8\), \(y=0\); \(x=-8\), \(y = 4\). So the correct pattern is that for every increase of 4 in \(x\), \(y\) decreases by 1.

So for \(x=-6\): \(x=-8 + 2\). When \(x=-8\), \(y = 4\). For an increase of 2 in \(x\) (from \(x=-8\) to \(x=-6\)), \(y\) decreases by \(\frac{2}{4}=\frac{1}{2}\). So \(y = 4-\frac{1}{2}=3.5\) (which is \(\frac{7}{2}\)).

For \(x=-3\): \(x=-4 + 1\). When \(x=-4\), \(y = 3\). For an increase of 1 in \(x\) (from \(x=-4\) to \(x=-3\)), \(y\) decreases by \(\frac{1}{4}\). So \(y=3-\frac{1}{4}=\frac{12 - 1}{4}=\frac{11}{4}=2.75\)? No, wait, when \(x=-4\), \(y = 3\); \(x = 0\), \(y = 2\) (difference of 4 in \(x\), difference of - 1 in \(y\)). So the equation is \(y=-\frac{1}{4}x + 2\). Let's substitute \(x=-3\):

\(y=-\frac{1}{4}\times(-3)+2=\frac{3}{4}+2=\frac{3 + 8}{4}=\frac{11}{4}=2.75\)? But looking at the graph, when \(x=-3\), the \(y\) - value should be \(2.75\)? Wait, maybe we made a mistake in the initial point selection. Let's use the two - point formula again. Let's take \((0,2)\) and \((8,0)\). The equation is \(y=-\frac{1}{4}x + 2\).

For \(x = 0\): \(y=-\frac{1}{4}(0)+2=2\)

For \(x = 3\): Substitute \(x = 3\) into \(y=-\frac{1}{4}x + 2\). \(y=-\frac{3}{4}+2=\frac{-3 + 8}{4}=\frac{5}{4}=1.25\)

For \(x = 6\): Substitute \(x = 6\) into \(y=-\frac{1}{4}x + 2\). \(y=-\frac{6}{4}+2=-\frac{3}{2}+2=\frac{-3 + 4}{2}=\frac{1}{2}=0.5\)

Wait, let's correct the earlier mistake. Let's re - establish the equation. Let's take two points: \((- 8,4)\) and \((8,0)\). The slope \(m=\frac{0 - 4}{8-(-8)}=\frac{-4}{16}=-\frac{1}{4}\). The equation using point - slope form with \((8,0)\) is \(y-0=-\frac{1}{4}(x - 8)\), which simplifies to \(y=-\frac{1}{4}x+2\), which is correct.

Now, let's fill in the table:

- When \(x=-6\): \(y=-\frac{1}{4}\times(-6)+2=\frac{6}{4}+2=\frac{3}{2}+2=\frac{3 + 4}{2}=\frac{7}{2}=3.5\)
- When \(x=-3\): \(y=-\frac{1}{4}\times(-3)+2=\frac{3}{4}+2=\frac{3+8}{4}=\frac{11}{4}=2.75\)? No, wait, when \(x=-4\), \(y = 3\) (from the graph: when \(x=-4\), moving up from the \(x\) - axis, \(y = 3\)). Let's check with \(x=-4\) in the equation: \(y=-\frac{1}{4}\times(-4)+2=1 + 2=3\), which matches. So when \(x=-3\), \(x=-4 + 1\), so \(y=3-\frac{1}{4}=\frac{12 - 1}{4}=\frac{11}{4}=2.75\) (or \(2\frac{3}{4}\)).

- When \(x = 0\): \(y=-\frac{1}{4}\times0+2=2\)
- When \(x = 3\): \(y=-\frac{1}{4}\times3+2=-\frac{3}{4}+2=\frac{-3 + 8}{4}=\frac{5}{4}=1.25\) (or \(1\frac{1}{4}\))
- When \(x = 6\): \(y=-\frac{1}{4}\times6+2=-\frac{3}{2}+2=\frac{-3 + 4}{2}=\frac{1}{2}=0.5\) (or \(\frac{1}{2}\))

Wait, but maybe the graph has a slope of \(-\frac{1}{4}\) and the \(y\) - intercept at 2. Let's re - check the values:

| \(x\) | \(y\) |
|-------|-------|
| \(-6\) | \(y =-\frac{1}{4}\times(-6)+2=\frac{6}{4}+2 = 1.5 + 2=3.5\) (or \(\frac{7}{2}\)) |
| \(-3\) | \(y=-\frac{1}{4}\times(-3)+2=\frac{3}{4}+2 = 0.75+2 = 2.75\) (or \(\frac{11}{4}\)) |
| \(0\) | \(y = 2\) |
| \(3\) | \(y=-\frac{1}{4}\times3+2=-0.75 + 2 = 1.25\) (or \(\frac{5}{4}\)) |
| \(6\) | \(y=-\frac{1}{4}\times6+2=-1.5 + 2 = 0.5\) (or \(\frac{1}{2}\)) |

But maybe we made a mistake in the slope. Let's look at the graph again. The line goes from \((- 8,4)\) to \((0,2)\) to \((8,0)\). So the change in \(x\) from \(-8\) to \(0\) is \(+8\), change in \(y\) is \(-2\), so slope is \(\frac{-2}{8}=-\frac{1}{4}\), correct. So the calculations are correct.

Answer:

\(x\)	\(y\)
\(-3\)	\(\frac{11}{4}\) (or \(2.75\))
\(0\)	\(2\)
\(3\)	\(\frac{5}{4}\) (or \(1.25\))
\(6\)	\(\frac{1}{2}\) (or \(0.5\))

If we want to write the answers as fractions:

For \(x=-6\), \(y = \frac{7}{2}\); for \(x=-3\), \(y=\frac{11}{4}\); for \(x = 0\), \(y = 2\); for \(x = 3\), \(y=\frac{5}{4}\); for \(x = 6\), \(y=\frac{1}{2}\)