Problem 44 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=-1.6 x+13.6$$

Step-by-Step Solution

Answer

The y-intercept is (0, 13.6) and the x-intercept is (8.5, 0).

Step 1: Identify the slope and y-intercept

The given equation is in the slope-intercept form, which is written as \(y=mx+c\). In this form, \(m\) is the slope and \(c\) is the y-intercept. For the equation \(y=-1.6x+13.6\), the slope \(m\) is -1.6 and the y-intercept \(c\) is 13.6.

Step 2: Plot the y-intercept

To find the y-intercept, set \(x=0\). Substituting 0 for \(x\), \(y = -1.6(0) + 13.6 = 13.6\). So, the y-intercept is (0, 13.6). Plot this point on the graph.

Step 3: Find the x-intercept

To find the x-intercept, set \(y=0\). Solve the equation \(0 = -1.6x + 13.6\). Rearrange to find \(x\): \(1.6x = 13.6\) \(x = \frac{13.6}{1.6} = 8.5\). So, the x-intercept is at (8.5, 0). Plot this point on the graph.

Step 4: Draw the line

Using the points (0, 13.6) and (8.5, 0) that have been plotted, draw a straight line through these points. This line represents the graph of the equation \(y = -1.6x + 13.6\).

Step 5: Label the intercepts

Clearly label the intercepts on the graph: (0, 13.6) for the y-intercept and (8.5, 0) for the x-intercept.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a way to write the equation of a straight line. It is given by the formula: \( y = mx + c \)
where:

• \( m \) is the slope of the line
• \( c \) is the y-intercept – the point where the line crosses the y-axis

This form makes it easy to identify both the slope and the y-intercept directly from the equation.
For the equation provided in the exercise, \( y = -1.6x + 13.6 \), the slope \( m \) is \( -1.6 \) and the y-intercept \( c \) is \( 13.6 \).
Knowing these values allows us to graph the line quickly.

x-intercept

The x-intercept is the point where the graph of an equation crosses the x-axis. This happens when the y-value is zero. To find the x-intercept of the equation:
\( y = -1.6x + 13.6 \)
we set \( y = 0 \).

• Start with the equation: \( 0 = -1.6x + 13.6 \)
• Rearrange the equation to solve for \( x \):
• \( 1.6x = 13.6 \)
• Divide both sides by 1.6:
• \( x = \frac{13.6}{1.6} = 8.5 \)

So, the x-intercept is at the point \( (8.5, 0) \). This means that the line crosses the x-axis at \( x = 8.5 \).
Plot this point on your graph to help visualize the line.

y-intercept

The y-intercept is the point where the graph crosses the y-axis, which happens when the x-value is zero. For the given equation, \( y = -1.6x + 13.6 \), finding the y-intercept involves setting \( x = 0 \).

• Substitute \( 0 \) for \( x \) in the equation:
• \( y = -1.6(0) + 13.6 = 13.6 \)

So, the y-intercept is at the point \( (0, 13.6) \).
This point is where the line crosses the y-axis. By plotting this point on a graph, you get an important reference for drawing the line.
Remember to label this intercept on your graph to show where the line meets the y-axis.