Question

which ordered pairs are solutions to the equation? select all that apply.
(5, -2) (-5, -6) (-3, 7)
(6, 4) (3, -5) (-6, 3)

Answer:

To determine which ordered pairs are solutions, we first find the equation of the line. The line passes through the origin \((0,0)\) and has a slope \(m\). Let's calculate the slope using two points, e.g., \((1,1)\) (since when \(x = 1\), \(y = 1\)):
\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1 - 0}{1 - 0}=1\).
So the equation is \(y = x\) (since \(b = 0\) from \(y = mx + b\)).

Now we check each ordered pair \((x,y)\) to see if \(y = x\):

• \((5, -2)\): \( -2

eq5\) → Not a solution.

• \((-5, -6)\): \( -6

eq -5\) → Not a solution.

• \((-3, 7)\): \(7

eq -3\) → Not a solution.

• \((6, 4)\): \(4

eq6\) → Not a solution.

• \((3, -5)\): \( -5

eq3\) → Not a solution.

• \((-6, 3)\): \(3

eq -6\) → Not a solution.

Wait, there seems to be a mistake. Let's re - evaluate the slope. Let's take two points from the graph. For example, when \(x = 2\), \(y = 2\); when \(x=-2\), \(y = - 2\). So the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2-(-2)}{2 - (-2)}=\frac{4}{4}=1\). So the equation is \(y=x\).

Now, let's check the correct solutions (where \(y = x\)):

• If we consider the graph, the line \(y = x\) has points where \(x\) and \(y\) are equal. But the given options have errors. Wait, maybe I misread the graph. Let's look again. The line passes through \((0,0)\) and \((2,2)\), \((-2,-2)\), etc. So the correct solutions should be pairs where \(y=x\). But none of the given options (as checked) satisfy \(y = x\). However, maybe the slope is different. Let's recalculate the slope with another pair. Let's take \((1,1)\) and \((2,2)\): slope is \(1\). So the equation is \(y=x\).

But the original checked options are incorrect. However, if we assume a different slope. Let's take two points: \((0,0)\) and \((1,1)\) (slope 1), \((2,2)\) (slope 1). So the correct solutions should be pairs where \(y = x\). But the given options don't have such pairs. There must be a mistake in the initial checking. Wait, maybe the line is \(y=x\), so the correct ordered pairs are those where \(x=y\). But in the given options, none of \((5, - 2)\), \((-5,-6)\), \((-3,7)\), \((6,4)\), \((3,-5)\), \((-6,3)\) have \(x = y\). So there is an error in the marked options. But if we re - examine the graph, maybe the slope is \(1\), so the correct solutions are pairs like \((1,1)\), \((2,2)\), \((-1,-1)\), etc. But since the problem asks to select from the given options, and all the marked options are incorrect (as per \(y = x\)), there is a mistake. However, if we consider a different equation. Let's calculate the slope between \((0,0)\) and \((5,6)\): slope \(m=\frac{6 - 0}{5 - 0}=\frac{6}{5}\), but that doesn't match. Alternatively, maybe the line is \(y=x + 0\), so the correct answer should be that none of the marked options are correct. But since the problem has marked options, maybe I made a mistake.

Wait, let's check the graph again. The line passes through \((0,0)\) and when \(x = 5\), \(y = 6\)? No, the graph shows that when \(x = 5\), \(y = 6\)? Wait, the y - axis at \(x = 5\) is at \(y = 6\)? Wait, maybe the grid is different. Let's count the grid squares. From \((0,0)\) to \((1,1)\) is one square, but maybe the slope is \(1\). I think there is a mistake in the problem's marked options. But if we follow the equation \(y=x\), the correct ordered pairs should have \(x=y\). Since none of the given options have \(x=y\), the correct answer is that none of the marked options are solutions. But since the problem expects some answers, maybe there is a miscalculation.

Wait, maybe the equation is \(y=x\), so the correct solutions are pairs where \(x = y\). But in the given options, there are no such pairs. So the answer is that none of the provided ordered pairs \((5, - 2)\), \((-5,-6)\), \((-3,7)\), \((6,4)\), \((3,-5)\), \((-6,3)\) are solutions to \(y=x\).