Problem 38 - Algebra 1

Question

Graph the line that has the given intercepts. $$ \begin{array}{l} x \text { -intercept: }-3 \\ y \text { -intercept: }-7 \end{array} $$

Step-by-Step Solution

Answer

The line crossing the x-axis at (-3, 0) and the y-axis at (0, -7) is the required graph.

Step 1: Plot the x-intercept

The x-intercept is given as -3. This intercept means that the line crosses the x-axis at the point (-3, 0). Start by plotting this point on a graph.

Step 2: Plot the y-intercept

The y-intercept is given as -7. This intercept means that the line crosses the y-axis at the point (0, -7). Now plot this point on the same graph.

Step 3: Draw the line

Now that we have the two intercepts, we can draw a line that passes through these two points. This line is the graph of the equation.

Key Concepts

Understanding the x-intercept

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 situated on the x-axis itself. Essentially, the x-intercept gives you an idea of the horizontal crossing point of the line.

To find the x-intercept, you need any linear equation in the form of \(ax + by = c\). You simply set \(y = 0\) and solve for \(x\). By doing so, you effectively remove the impact of any vertical movement (y-axis influence) and find where the line touches the x-axis.

• For example, with the x-intercept at -3, the line crosses the x-axis at the point \((-3, 0)\).
• This means the line moves 3 units to the left of the origin before reaching the x-axis.

Plotting the x-intercept is straightforward. Simply start at the origin (\(0, 0\)) and move horizontally along the x-axis to the point \((-3, 0)\). Mark this point. This point serves as one of your guides in sketching the line representing your equation.

Unpacking the y-intercept

The y-intercept is the location where the line crosses the y-axis, meaning this point always has an \(x\)-coordinate of zero. The y-intercept helps to describe how far along the vertical axis the line meets zero on the horizontal axis.
To determine the y-intercept for a given linear equation \(ax + by = c\), you set \(x = 0\) and solve for \(y\). This action effectively zeros out any lateral movement, pinpointing where the line intersects with the y-axis.

• In the exercise given, the y-intercept is -7, indicating a crossing at \((0, -7)\).
• This means the line runs 7 units below the origin on the y-axis.

To plot the y-intercept, position yourself at the origin \((0, 0)\) and move down along the y-axis to \((0, -7)\). Mark this critical point, which serves as a guide for the path of your line.

Learning to plot points

Plotting points is a fundamental skill in graphing linear equations. It involves marking specific points on the Cartesian plane based on coordinates provided or calculated from your equation. These plotted points act as reference markers to visualize and draw straight lines or curves corresponding to equations.
When you graph a line using its intercepts:

• Begin by identifying the x-intercept and y-intercept using the steps above.
• Place these two critical points, \((-3, 0)\) and \((0, -7)\), on your graph.

Once you plot these points, use a ruler or a straight edge to draw a line that smoothly passes through both points. This basic action of connecting two dots might seem simple, but it represents the complete path of a linear function.
The clarity and simplicity of plotting points not only facilitates understanding but sets a foundation for more complex graphing challenges. It empowers you to visualize how mathematical relationships manifest on the plane, making functions more accessible and intuitive.