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The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero because the line is touching or crossing the x-axis at this "height." To find the x-intercept, you simply set the equation equal to zero for y and solve for x. For example, in the equation given, which is already in slope-intercept form, you have to substitute zero for y: \[ 0 = 3x + 6 \]To isolate x, you start by subtracting 6 from both sides:\[ -6 = 3x \]Then, you divide both sides by 3 to solve for x: \[ x = -2 \]So, the x-intercept is at the point \((-2, 0)\). This means the line crosses the x-axis where the x value is -2.

The y-intercept is the point where the line crosses the y-axis, and at this moment, the x-coordinate is zero. This is often the starting point when you are plotting a line on a graph, as it is straightforward to identify and calculate.In a line equation in the slope-intercept form \( y = mx + b \), the y-intercept is directly given as \( b \). For our specific line, the equation is:\[ y = 3x + 6 \]Here, you can immediately see that the y-intercept \( b \) is 6, indicating that the line crosses the y-axis at the point \( (0, 6) \). This means the line touches or crosses the y-axis 6 units above where the axis meets the origin.

The slope-intercept form is a very common way to write the equation of a straight line. It is represented as \( y = mx + b \), where:- \( m \) is the slope of the line.- \( b \) is the y-intercept.The slope \( m \) describes how steep the line is. Practically, it indicates how much y increases for a one-unit increase in x. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. Inour given equation \( y = 3x + 6 \), the slope is 3. This tells us that for every unit increase along the x-axis, the line will rise 3 units. Additionally, since \( b \) (the y-intercept) is 6, we know that when x is zero, y will be 6. This layout makes it easy to graph, even without a calculator, because you only need the slope and y-intercept to begin plotting the line.

Graphically representing a straight line helps in better understanding how changes in the equation affect the line’s path and position on the graph.When graphing from the slope-intercept form such as \( y = 3x + 6 \), you can start by locating the y-intercept on the graph. Plot the point \( (0, 6) \) on the y-axis. Next, use the slope to find another point. From \( (0, 6) \), apply the slope of 3, which means you go up 3 units and 1 unit right to get to the point \( (1, 9) \).Another helpful point is the x-intercept, calculated earlier as \((-2, 0)\). Plotting these points helps ensure the accuracy of the line.Connect these dots with a straight line extending beyond them.

• Start at the y-intercept: \( (0, 6) \).
• Move according to the slope: 3 units up and 1 unit right.
• Confirm with the x-intercept: \( (-2, 0) \).

This process results in an accurate representation of the line described by the equation, reinforcing your understanding of intercepts and slope-intercept form. Checking the drawn line with a calculator can further clarify any doubts.