Problem 34 - Algebra 1: Concepts and Skills

Question

Graph the line that has the given intercepts. $$ \begin{array}{l} {x \text { -intercept: }4} \\ {y \text { -intercept: }6} \end{array} $$

Step-by-Step Solution

Answer

To graph a line with x-intercept 4 and y-intercept 6, plot the points (4, 0) and (0, 6) on the graph and draw a line through them.

Step 1: Plotting the x-intercept

Plot the x-intercept on the x-axis. In this case, the x-intercept is 4. In order to plot this, find 4 on the x-axis and mark this point. The y-coordinate of the x-intercept is always 0, so this point is (4, 0).

Step 2: Plotting the y-intercept

Next, plot the y-intercept on the y-axis. The y-intercept is 6 for this exercise. Locate 6 on the y-axis and mark the point. Remember, the x-coordinate for a y-intercept is always 0, which makes this point (0, 6).

Step 3: Drawing the line

Finally, draw a straight line that passes through both points. This line represents the linear equation with the given intercepts.

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial concept when graphing linear equations. It represents the point where the graph of a line crosses the x-axis. At this point, the value of the y-coordinate is zero.
To find the x-intercept, you simply set the y-value to zero in the linear equation and solve for x. This will give you the exact location of the x-intercept on the x-axis.
For example, if the x-intercept is given as 4, the corresponding point on the graph is (4, 0). This means that you will mark a point at number 4 on the x-axis.
Understanding this concept allows you to quickly identify where the line passes through the x-axis, aiding in the accurate sketching of the graph.
Things to remember about the x-intercept:

• Occurs on the x-axis where y is zero.
• Calculate by solving the equation with y = 0.
• Is written as a point (x, 0).

Exploring the Y-Intercept

The y-intercept is a key element in graphing linear equations, indicating where the line crosses the y-axis. At this intersection, the x-coordinate is always zero.
To find the y-intercept, you usually set the x-value to zero in the equation and solve for y. This gives you the point on the y-axis where the intercept occurs.
In the example provided, the y-intercept is 6. Thus, the plotted point would be (0, 6) on the graph's y-axis. This is a simple yet powerful part of understanding linear equations, as it helps to determine the shape and position of the line.
Key points about the y-intercept:

• Occurs on the y-axis where x is zero.
• Find it by solving the equation with x = 0.
• It is expressed as the point (0, y).

Plotting Points and Drawing the Line

Once you have determined both the x-intercept and the y-intercept, you are ready to plot these points on the graph. Start by marking the x-intercept on the x-axis and then the y-intercept on the y-axis.
These points are essential indicators of the line’s path, providing the framework needed to draw it accurately.
Once both points are plotted, take a ruler or straight edge and connect them to form the complete line. This visual representation is your linear equation.
The process of plotting points involves:

• Locating the x-intercept and placing it on the x-axis.
• Identifying the y-intercept and marking it on the y-axis.
• Using a straight edge to connect the dots and form a line.

This step not only helps visualize the equation but also demonstrates the relationship between the intercepts and the overall equation. By plotting these points and drawing a line through them, you bring the equation to life, portraying how it behaves on a graph.