Problem 28 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

x-intercept is (4, 0); y-intercept is (0, 4). Line passes through these points.

Step 1: Identify the Standard Form Coefficients

The equation given is a linear equation in the standard form, which is written as \(Ax + By = C\). In this equation, \(A = 1\), \(B = 1\), and \(C = 4\).

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y\) to zero and solve for \(x\). This gives us the point where the line intersects the \(x\)-axis.\[x + 0 = 4 \x = 4\]The \(x\)-intercept is \((4, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x\) to zero and solve for \(y\). This gives us the point where the line intersects the \(y\)-axis.\[0 + y = 4 \y = 4\]The \(y\)-intercept is \((0, 4)\).

Step 4: Sketch the Graph

To graph the line, plot the intercepts on the Cartesian plane. Plot \((4, 0)\) for the \(x\)-intercept and \((0, 4)\) for the \(y\)-intercept. Draw a straight line through these two points to represent the equation \(x + y = 4\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a line is the point where the graph intersects the x-axis. At this point, the value of y is always zero because the point lies directly on the horizontal axis. To find the x-intercept from a linear equation, you simply set y to zero and solve for x.

For example, in the equation \(x + y = 4\), substituting \(y = 0\) gives us \(x + 0 = 4\). So, \(x = 4\). This means our line crosses the x-axis at the point \((4, 0)\).

• **Key Insight**: The x-intercept helps you pinpoint one of the two critical anchor points needed to graph a linear equation.
• **Quick Tip**: Remember, when finding the x-intercept, y is always zero.

Exploring the Y-Intercept

Much like its x-counterpart, the y-intercept is where the line crosses the y-axis. Here, the x-value is automatically set to zero because the crossing occurs right along the vertical axis. To pinpoint the y-intercept, you solve the linear equation with x set to zero.

Using our equation \(x + y = 4\), we substitute \(x = 0\), resulting in \(0 + y = 4\). Hence, \(y = 4\), indicating the y-intercept is at \((0, 4)\).

• **Key Insight**: Identifying the y-intercept is crucial for setting up your graph, as it provides another necessary fixed point.
• **Quick Tip**: The x-value is always zero at the y-intercept. Remember this to find it effortlessly!

Mastering Graphing Lines

Graphing lines becomes straightforward once you have the x- and y-intercepts. These two points will guide the drawing of the line.

Here's how to graph starting from the intercepts:

• Start by plotting the x-intercept, which you found at \((4, 0)\).
• Next, plot the y-intercept at \((0, 4)\).
• Use a ruler or draw a line straight through these two points. You can extend the line in both directions, ensuring it’s straight.

Voila! You have successfully graphed the line \(x + y = 4\) using just its intercepts. This method is efficient and effective, especially with straightforward equations. Always ensure your graph is neat for accuracy and clarity.

• **Key Insight**: Graphing via intercepts is a tangible way to visualize a linear equation and is foundational for understanding the relationship between algebra and geometry.
• **Quick Tip**: Connecting two fixed points guarantees a straight line, a hallmark of linear equations.