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The x-intercept of a line is the point where the line crosses the x-axis. To find this point, you need to set the y-value to zero since any point on the x-axis has a y-value of zero. For the equation given in the exercise, which is \[3x - y = 0,\]we substitute y with zero and solve the resulting equation:\[3x - 0 = 0.\]From this equation, we immediately find that:\[3x = 0,\]which simplifies to:\[x = 0.\]So, the x-intercept is at the origin, the point \((0, 0)\). This means the graph of the line touches the x-axis at this particular point.

The y-intercept of a line is the point where the line crosses the y-axis. To identify the y-intercept, we set the x-value to zero in the equation because points on the y-axis have an x-value of zero. Using the line equation from the exercise: \[3x - y = 0,\]we substitute x with zero:\[3(0) - y = 0.\]This simplifies to:\[-y = 0.\]Solving this, we find:\[y = 0.\]Thus, the y-intercept is also at the origin, \((0, 0)\). This indicates that the graph intersects the y-axis at this point.

The slope-intercept form of a linear equation is an essential concept in algebra. It is typically written as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its steepness.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

In the exercise, the equation \(3x - y = 0\) can be rearranged into slope-intercept form:\[y = 3x.\]Here, the slope \(m\) is 3, which means for every unit increase in \(x\), \(y\) increases by 3 units. The y-intercept \(b\) is 0, meaning the line passes through the origin, \((0, 0)\). This form is particularly useful for quickly identifying the slope and y-intercept of a line.

Graphing linear equations involves drawing a line that best represents the equation on a coordinate plane. To graph an equation like \(3x - y = 0\), follow these steps:1. **Find the Intercepts**: We found both the x-intercept and y-intercept to be \((0, 0)\).
2. **Write in Slope-Intercept Form**: Rewrite the equation as \(y = 3x\) for easier graphing.
3. **Draw the Line**: Since the line passes through the origin and has a slope of 3, plot the origin and choose another point by using the slope. From \((0, 0)\), move 1 unit right and 3 units up to point \((1, 3)\).
4. **Extend the Line**: Connect these points and extend the line in both directions.
This creates a visual representation of the equation where you can clearly see the direction and steepness of the line through the origin.