Problem 32 - Thomas Calculus

Question

In Exercises \(31-34,\) find the line's \(x\) - and \(y\) -intercepts and use this information to graph the line. $$ x+2 y=-4 $$

Step-by-Step Solution

Answer

The x-intercept is (-4, 0) and the y-intercept is (0, -2).

Step 1: Identify Equation Form

We start with the linear equation given: \( x + 2y = -4 \). This is in standard form \( Ax + By = C \), where \( A = 1 \), \( B = 2 \), and \( C = -4 \).

Step 2: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( x + 2y = -4 \). This results in \( x + 2(0) = -4 \), simplifying to \( x = -4 \). Thus, the x-intercept is \( (-4, 0) \).

Step 3: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( x + 2y = -4 \). This results in \( 0 + 2y = -4 \), or \( 2y = -4 \). Solving for \( y \), we divide both sides by 2, yielding \( y = -2 \). Hence, the y-intercept is \( (0, -2) \).

Step 4: Plot the Intercepts

Plot the intercepts found: the x-intercept \( (-4, 0) \) and the y-intercept \( (0, -2) \) on the Cartesian coordinate plane.

Step 5: Draw the Line

Draw a straight line connecting the two plotted intercept points. This is the graph of the line represented by the equation \( x + 2y = -4 \).

Key Concepts

Understanding X-Intercepts

An x-intercept is where a line crosses the x-axis. At this point, the value of y is always zero. You can find the x-intercept of an equation by substituting zero for y and solving for x. This gives you the precise point where the line meets the x-axis.
In our example equation, \( x + 2y = -4 \), setting \( y = 0 \) leads to \( x = -4 \). Thus, the x-intercept is \((-4, 0)\).

Understanding x-intercepts helps in graphing, because it gives one of the key locations through which the line passes. This is crucial for an accurate graph.

• Set \( y = 0 \) in the equation to find the x-intercept.
• Solve the resulting equation for x to get the intercept point.
• Interpret the intercept as a coordinate \( (x, 0) \).

Understanding Y-Intercepts

A y-intercept is the point where a line crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept, you need to set x to zero in the equation and solve for y. This tells you exactly where the line meets the y-axis.
Using the same example, \( x + 2y = -4 \), by setting \( x = 0 \), we find that \( y = -2 \). This makes the y-intercept \((0, -2)\).

Knowing the y-intercept is essential for graphing a line because it gives another vital point to accurately place the line on the graph.

• Set \( x = 0 \) in the equation to determine the y-intercept.
• Solve for y to find the corresponding intercept point.
• Understand this intercept as a coordinate \( (0, y) \).

Graphing Lines Using Intercepts

Graphing a line using its x- and y-intercepts is straightforward and efficient. To graph a line, these intercepts are plotted on a coordinate plane to provide clear guidance on the line's path.
For the equation \( x + 2y = -4 \), you've already found two crucial points: the x-intercept \((-4, 0)\) and the y-intercept \((0, -2)\). These points help form the backbone of the line.

To graph:

• First, plot the x-intercept \((-4, 0)\) on the graph.
• Next, plot the y-intercept \((0, -2)\).
• Finally, draw a straight line connecting these two points.

Using intercepts simplifies the graphing process, minimizing errors and ensuring accuracy.