Problem 45 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=24 x+444$$

Step-by-Step Solution

Answer

Y-intercept: (0, 444); X-intercept: (-18.5, 0)

Step 1: Identify the equation

The given equation is in the slope-intercept form: $$y = 24x + 444$$. Here, the slope ( m ) is 24 and the y-intercept ( b ) is 444.

Step 2: Determine the y-intercept

To find the y-intercept, set $$x = 0$$. This gives you: $$y = 24(0) + 444 = 444$$. So, the y-intercept is at the point (0, 444).

Step 3: Determine the x-intercept

To find the x-intercept, set $$y = 0$$. Solve for $$x$$ in the equation: $$0 = 24x + 444$$. Subtract 444 from both sides: $$-444 = 24x$$. Divide by 24: $$x = -18.5$$. So, the x-intercept is at the point (-18.5, 0).

Step 4: Plot the intercepts

Plot the points (0, 444) and (-18.5, 0) on a coordinate plane as the y-intercept and x-intercept, respectively.

Step 5: Draw the line

Using the two intercept points, draw a straight line through them extending in both directions to represent the graph of the equation $$y = 24x + 444$$.

Step 6: Label the intercepts

Clearly label the intercepts on the graph: (0, 444) for the y-intercept and (-18.5, 0) for the x-intercept.

Key Concepts

Slope-Intercept Form

Understanding how to work with the slope-intercept form is crucial for graphing linear equations. The slope-intercept form of a linear equation is given by \(y = mx + b\), where

• \(m\) is the slope of the line, showing how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

. For the equation \(y=24x+444\), the slope \(m\) is 24, which means for every 1 unit increase in \(x\), \(y\) increases by 24 units. The y-intercept \(b\) is 444, showing that the line crosses the y-axis at \(y=444\).

x-intercept Calculation

To find the x-intercept, we need to determine where the line crosses the x-axis. This means we set \(y = 0\) in the equation and solve for \(x\). Using the given equation \(y=24x+444\):

1. Set \(y = 0\): \(0 = 24x + 444\).
2. Solve the equation: subtract 444 from both sides to get \(-444 = 24x\).
3. Divide both sides by 24 to get \(x = -18.5\).

Hence, the x-intercept is at the point (-18.5, 0).

y-intercept Calculation

Finding the y-intercept is simpler. We need to find where the line crosses the y-axis, which happens when \(x = 0\). Using the equation \(y=24x+444\):

• Set \(x = 0\): \(y = 24(0) + 444\).
• This means \(y = 444\).

Thus, the y-intercept is at the point (0, 444). This point tells us where the line will touch the vertical axis.

Plotting Points

Plotting points is essential for accurately drawing the graph of a linear equation. For this equation, we have two key points: the y-intercept (0, 444) and the x-intercept (-18.5, 0).

• First, mark the point (0, 444) on the y-axis.
• Next, mark the point (-18.5, 0) on the x-axis.

Once these points are marked, draw a straight line through them. This line represents the equation \(y = 24x + 444\).
Make sure to extend the line in both directions and label the intercepts clearly to complete the graph.