Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-2,-7) \quad m=0\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -7\).

Step 1: Identify the Given

The problem provides the slope \(m = 0\) and the coordinates of a point the line goes through, \((-2, -7)\).

Step 2: Apply the Slope-Intercept Form

In the slope-intercept form equation of a line \(y = mx + b\), sub in the given slope \(m = 0\) and the known y-value from the point (which is -7) to get -7 = 0x + b. This simplifies to -7 = b.

Step 3: Write the Line Equation

Now that the slope \(m\) and the y-intercept \(b\) are known, substitute them into the equation \(y = mx + b\) to get \(y = 0x - 7\). The term with x will disappear because anything multiplied by 0 is 0, leaving us with the equation of the line: \(y = -7\).

Step 4: Sketch the Line

Sketching this line, it appears as a horizontal line passing through the y-coordinates at -7.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a standard way to write the equation of a line, given by the formula \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it provides a quick way to see how steep the line is (the slope) and where it starts on the y-axis.

In our exercise, we are given a slope \(m = 0\) and a point that the line passes through, \((-2, -7)\). Substitute \(m = 0\) and the y-value from the point into the equation to find the y-intercept. When the slope is zero, the line is horizontal and runs parallel to the x-axis, so the equation becomes simplified: the \(x\) term disappears, resulting in \(y = b\).

For our line, after substituting, we find the equation \(y = -7\), indicating a constant y-value across all x-values. This shows why understanding the slope-intercept form is important—it's quickly recognizable that this line is indeed horizontal.

Point-Slope Formula

The point-slope formula is another form of writing the equation of a line, especially useful when a point on the line and the slope are known. It is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.

In our case, the point provided is \((-2, -7)\) and the slope \(m = 0\). Plugging these into the formula, we have:
\[y - (-7) = 0(x - (-2))\]
Simplifying, this becomes \(y + 7 = 0\), and further simplifying, \(y = -7\), which matches what we found earlier using the slope-intercept form.

The beauty of the point-slope formula is that it adapts well for finding equations when a specific point is known. Although it concurs with the prior method, it's another tool in your mathematical toolbox.

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the equation and drawing the line through them. Understanding the slope-intercept form and point-slope formula makes this process straightforward.

For the equation \(y = -7\), no matter what x-value you choose, the y-value will always be -7. This tells us that every point on the line shares the same y-coordinate, making it a horizontal line. To graph it:

• Draw the y-axis and x-axis on a graph.
• Locate the y-intercept at \(y = -7\) and make a mark.
• Draw a straight line through the mark parallel to the x-axis, since the slope is zero and does not rise or fall with x.

This consistency forms the visual representation of our linear equation.
Whether in an algebra class or practical real-world graphing, visualizing linear equations helps reinforce the concepts of slope and intercept.