Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Graph each line by hand. Give the \(x\) - and y-intercepts. $$x+y=4$$

Step-by-Step Solution

Answer

The x-intercept is (4, 0) and the y-intercept is (0, 4). Plot these points and draw the line.

Step 1: Rearrange for y-intercept

To find the y-intercept, rearrange the given equation to make y the subject. Start with the equation: \(x+y=4\), and solve for y: \(y=4-x\). At the y-intercept, set \(x=0\). Substitute \(x=0\) into the equation: \(y=4-0=4\). Thus, the y-intercept is \( (0, 4) \).

Step 2: Rearrange for x-intercept

To find the x-intercept, set \(y=0\) in the original equation: \(x+y=4\). Substitute \(y=0\) into the equation: \(x+0=4\), which simplifies to \(x=4\). Thus, the x-intercept is \((4, 0)\).

Step 3: Plot the intercepts

Using the intercepts found, plot the points \( (0, 4) \) and \( (4, 0) \) on a graph. The y-intercept at \( (0, 4) \) is where the line crosses the y-axis, and the x-intercept at \( (4, 0) \) is where the line crosses the x-axis.

Step 4: Draw the line

Connect the plotted points \( (0, 4) \) and \( (4, 0) \) with a straight edge. Extend the line across the graph, ensuring that it passes through both intercepts, to accurately represent the line described by the equation \(x+y=4\).

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is a crucial point where the graph crosses the x-axis. To find this intercept for any equation, you simply set the value of y to zero and solve for x.

In the equation provided, \(x + y = 4\), we identify the x-intercept by substituting \(y = 0\) into the equation. This simplifies to \(x + 0 = 4\), which further reduces to \(x = 4\).

Therefore, the x-intercept is the point \((4, 0)\). Here, the graph touches or crosses the x-axis at the coordinate where x equals 4, and y is zero. This point is fundamental when plotting the line, as it guides us in shaping how the line should stretch across the graph.

Discovering the y-intercept

The y-intercept is another essential component of graphing linear equations. It is the point where the line crosses the y-axis. To discover the y-intercept, you set \(x = 0\) in the equation and solve for y.

For our equation \(x + y = 4\), setting \(x = 0\) leads to \(0 + y = 4\). Therefore, \(y = 4\).

Thus, the y-intercept is at the point \((0, 4)\). This point indicates where the graph intersects the y-axis, and is a vital reference point when drawing the line on the graph, setting the exact path it should take vertically.

The Art of Plotting Points

Plotting points on a graph is a straightforward yet essential skill in graphing linear equations accurately. It involves marking specific coordinates, like intercepts, which serve as a guide for sketching the line.

For the given equation \(x + y = 4\), we have identified two key points: the x-intercept \((4, 0)\) and the y-intercept \((0, 4)\).

To plot these points:

• Start with the x-intercept \((4, 0)\), place a point at where \(x = 4\) on the x-axis and \(y = 0\).
• Then, proceed to the y-intercept \((0, 4)\), marking a point where \(x = 0\) and \(y = 4\) on the y-axis.

Once these points are plotted, connecting them with a straight line will draw a clear representation of the equation.

This process not only visualizes the relationship described by the equation but also confirms the linear path determined by both intercepts.