The graph of linear equatio 2x-y=6 will pass through which quadrants(s)

Answer

To determine which quadrants the graph of the linear equation \(2x - y = 6\) passes through, we will follow these steps:

### Step 1: Rewrite the equation in slope-intercept form
To analyze the graph, we can rewrite the equation in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Starting with the original equation:
\[ 2x - y = 6 \]

We can isolate \(y\):
\[ -y = -2x + 6 \]
\[ y = 2x - 6 \]

### Step 2: Find the y-intercept
The y-intercept occurs when \(x = 0\). We can find the y-intercept by substituting \(x = 0\) into the equation:
\[ y = 2(0) - 6 = -6 \]
Thus, the y-intercept is at the point \((0, -6)\).

### Step 3: Find the x-intercept
The x-intercept occurs when \(y = 0\). We can find the x-intercept by substituting \(y = 0\) into the equation:
\[ 0 = 2x - 6 \]
\[ 2x = 6 \]
\[ x = 3 \]
Thus, the x-intercept is at the point \((3, 0)\).

### Step 4: Plot the points
Now we have two points to plot on the graph:
1. The y-intercept: \((0, -6)\)
2. The x-intercept: \((3, 0)\)

### Step 5: Draw the line
Using the two points, we can draw the line representing the equation \(2x - y = 6\).

### Step 6: Determine the quadrants
Now we analyze the line:
- The line crosses the y-axis at \((0, -6)\), which is in the fourth quadrant.
- The line crosses the x-axis at \((3, 0)\), which is in the first quadrant.

Since the line extends infinitely in both directions, we can determine which quadrants it passes through:
- The line will also extend into the third quadrant as it moves left from the y-intercept.

### Conclusion
The graph of the linear equation \(2x - y = 6\) passes through the **first**, **third**, and **fourth** quadrants.

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