Question

which of the following graphs matches the equation:
2x - 4y = 8

Explanation:

Step1: Convert to slope - intercept form

We start with the equation \(2x - 4y=8\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

First, subtract \(2x\) from both sides: \(- 4y=-2x + 8\).

Then, divide each term by \(-4\): \(y=\frac{-2x}{-4}+\frac{8}{-4}\), which simplifies to \(y=\frac{1}{2}x - 2\).

Step2: Analyze the slope and y - intercept

The slope \(m=\frac{1}{2}\) (positive, so the line should be increasing from left to right) and the \(y\) - intercept \(b=-2\) (the line crosses the \(y\) - axis at \((0, - 2)\)).

Now let's analyze each graph:
- The first graph has a negative slope, so we can eliminate it.
- The second graph (blue - boxed) has a \(y\) - intercept at \(y = 0\) (it crosses the \(y\) - axis at \((0,0)\)), so we can eliminate it.
- The third graph: The slope is positive (\(\frac{1}{2}\)) and the \(y\) - intercept is \(- 2\) (it crosses the \(y\) - axis at \((0,-2)\))? Wait, no, looking at the third graph, it seems to cross the \(y\) - axis at \((0, - 2)\)? Wait, no, let's check the fourth graph. Wait, maybe I made a mistake. Wait, let's re - check the third and fourth graphs.

Wait, the equation \(y=\frac{1}{2}x - 2\). Let's find the \(x\) - intercept by setting \(y = 0\): \(0=\frac{1}{2}x-2\), then \(\frac{1}{2}x=2\), so \(x = 4\). So the line should cross the \(x\) - axis at \((4,0)\) and the \(y\) - axis at \((0,-2)\).

Looking at the fourth graph: When \(x = 10\), \(y=\frac{1}{2}(10)-2=5 - 2 = 3\)? Wait, no. Wait, let's check the third graph. Wait, maybe the third graph (the one in the bottom - left, the third graph overall) has a \(y\) - intercept at \((0,-2)\) and an \(x\) - intercept at \((4,0)\)? Wait, no, the fourth graph: Let's calculate two points. When \(x = 0\), \(y=-2\); when \(y = 0\), \(x = 4\).

Looking at the four graphs:
- Graph 1: Negative slope, eliminate.
- Graph 2: \(y\) - intercept at \(0\), eliminate.
- Graph 3: Let's see, if we look at the grid, the line in graph 3 (bottom - left) seems to cross the \(y\) - axis at \((0,-2)\) and the \(x\) - axis at \((4,0)\)? Wait, no, the fourth graph (bottom - right): Let's check the \(x\) - intercept. If \(y = 0\), \(x = 4\). The fourth graph, when \(x = 10\), \(y=\frac{1}{2}(10)-2=3\), but the fourth graph's line at \(x = 10\) has \(y = 10\)? No, that can't be. Wait, maybe I misread the graphs. Wait, the equation is \(y=\frac{1}{2}x-2\). Let's list the points:

When \(x = 0\), \(y=-2\); when \(x = 4\), \(y = 0\); when \(x=-4\), \(y=-4\).

Now, looking at the fourth graph (the one at the bottom - right), the line passes through points that when \(x = 0\), \(y=-2\) and when \(x = 10\), \(y=\frac{1}{2}(10)-2 = 3\)? No, the fourth graph's line at \(x = 10\) is at \(y = 10\)? That doesn't match. Wait, the third graph (bottom - left): Let's see, the line in the third graph (bottom - left) has a slope of \(\frac{1}{2}\) and \(y\) - intercept of \(-2\)? Wait, no, the third graph (bottom - left) seems to cross the \(y\) - axis at \((0,-2)\) and the \(x\) - axis at \((4,0)\). Wait, maybe the correct graph is the bottom - left graph (the third graph overall). Wait, maybe I made a mistake in the initial analysis.

Wait, let's re - express the equation \(2x-4y = 8\). Let's find two points:

When \(x = 0\), \(-4y=8\), so \(y=-2\). So the point is \((0,-2)\).

When \(y = 0\), \(2x=8\), so \(x = 4\). So the point is \((4,0)\).

Now, let's check the graphs:

- Graph 1: Slope negative, wrong.
- Graph 2: Passes through \((0,0)\), wrong.
- Graph 3 (bottom - left): Let's see, does it pass through \((0,-2)\) and \((4,0)\)? From the grid, if we look at the bottom - left graph, when \(x = 0\), \(y=-2\) (since it crosses the \(y\) - axis at \((0,-2)\)) and when \(x = 4\), \(y = 0\) (crosses the \(x\) - axis at \((4,0)\))? Wait, no, the bottom - left graph's line seems to cross the \(y\) - axis at \((0,-2)\) and the \(x\) - axis at \((4,0)\)? Wait, the bottom - right graph: Let's check the \(x\) - intercept. If \(x = 10\), \(y=\frac{1}{2}(10)-2 = 3\), but the bottom - right graph's line at \(x = 10\) has \(y = 10\)? No, that's not right. Wait, maybe the correct graph is the bottom - left graph (the third graph, the one in the bottom - left corner). Wait, maybe I messed up the numbering. Let's number the graphs:

Top - left: Graph 1

Top - right: Graph 2 (blue - boxed)

Bottom - left: Graph 3

Bottom - right: Graph 4

Graph 1: Negative slope, eliminate.

Graph 2: \(y\) - intercept at \(y = 0\), eliminate.

Graph 3: Let's see, the line in Graph 3 has a slope of \(\frac{1}{2}\) (since for every 2 units we move to the right, we move up 1 unit) and \(y\) - intercept at \((0,-2)\). Let's check the \(x\) - intercept: when \(y = 0\), \(x = 4\). So if we look at Graph 3, when \(x = 4\), \(y = 0\), which matches.

Graph 4: Let's check the \(x\) - intercept. If \(y = 0\), \(x = 4\), but in Graph 4, the line at \(x = 10\) has \(y=\frac{1}{2}(10)-2=3\), but the line in Graph 4 seems to have a higher \(y\) - value at \(x = 10\), so it's not correct.

So the correct graph is the bottom - left graph (Graph 3). Wait, but let's re - check the equation. \(y=\frac{1}{2}x - 2\). The slope is positive, \(y\) - intercept at \((0,-2)\), \(x\) - intercept at \((4,0)\). The bottom - left graph (Graph 3) has these characteristics.

Answer:

The bottom - left graph (the third graph, the one in the bottom - left corner of the four - graph grid)