Question

which ordered pairs represent points on the graph of this equation? select all that apply.
$y = -\frac{3}{4}x - \frac{1}{2}$
$(-2, 6)$ $(0, -4)$ $(5, 0)$
$(2, -2)$ $(6, -5)$ $(-6, 4)$

Explanation:

To determine which ordered pairs \((x, y)\) lie on the graph of the equation \(y = -\frac{3}{4}x - \frac{1}{2}\), we substitute the \(x\)-value of each ordered pair into the equation and check if the resulting \(y\)-value matches the given \(y\)-value.

Step 1: Check \((-2, 6)\)

Substitute \(x = -2\) into the equation:
\[
y = -\frac{3}{4}(-2) - \frac{1}{2} = \frac{6}{4} - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1
\]
The calculated \(y\)-value is \(1\), but the given \(y\)-value is \(6\). So, \((-2, 6)\) is not on the graph.

Step 2: Check \((0, -4)\)

Substitute \(x = 0\) into the equation:
\[
y = -\frac{3}{4}(0) - \frac{1}{2} = 0 - \frac{1}{2} = -\frac{1}{2}
\]
The calculated \(y\)-value is \(-\frac{1}{2}\), but the given \(y\)-value is \(-4\). So, \((0, -4)\) is not on the graph.

Step 3: Check \((5, 0)\)

Substitute \(x = 5\) into the equation:
\[
y = -\frac{3}{4}(5) - \frac{1}{2} = -\frac{15}{4} - \frac{1}{2} = -\frac{15}{4} - \frac{2}{4} = -\frac{17}{4} = -4.25
\]
The calculated \(y\)-value is \(-4.25\), but the given \(y\)-value is \(0\). So, \((5, 0)\) is not on the graph.

Step 4: Check \((2, -2)\)

Substitute \(x = 2\) into the equation:
\[
y = -\frac{3}{4}(2) - \frac{1}{2} = -\frac{6}{4} - \frac{1}{2} = -\frac{3}{2} - \frac{1}{2} = -2
\]
The calculated \(y\)-value is \(-2\), which matches the given \(y\)-value. So, \((2, -2)\) is on the graph.

Step 5: Check \((6, -5)\)

Substitute \(x = 6\) into the equation:
\[
y = -\frac{3}{4}(6) - \frac{1}{2} = -\frac{18}{4} - \frac{1}{2} = -\frac{9}{2} - \frac{1}{2} = -5
\]
The calculated \(y\)-value is \(-5\), which matches the given \(y\)-value. So, \((6, -5)\) is on the graph.

Step 6: Check \((-6, 4)\)

Substitute \(x = -6\) into the equation:
\[
y = -\frac{3}{4}(-6) - \frac{1}{2} = \frac{18}{4} - \frac{1}{2} = \frac{9}{2} - \frac{1}{2} = 4
\]
The calculated \(y\)-value is \(4\), which matches the given \(y\)-value. So, \((-6, 4)\) is on the graph.

Answer:

The ordered pairs that represent points on the graph of the equation \(y = -\frac{3}{4}x - \frac{1}{2}\) are:
\((2, -2)\), \((6, -5)\), \((-6, 4)\)