Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Through \((-5,8),\) parallel to \(y=-0.2 x+6\)

Step-by-Step Solution

Answer

The equation is \( y = -0.2x + 7 \).

Step 1: Identify the Slope of the Given Line

The given equation of the line is in slope-intercept form, which is \( y = mx + c \). Here, \( m = -0.2 \), which is the slope.

Step 2: Use the Slope for Parallel Line

Parallel lines have the same slope. Therefore, the slope of the line we need to find is also \( m = -0.2 \).

Step 3: Use the Point-Slope Form

Use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \). Substitute the point \((-5, 8)\) and the slope \( m = -0.2 \) into this formula: \[ y - 8 = -0.2(x + 5) \]

Step 4: Simplify the Equation into Slope-Intercept Form

Distribute the slope on the right side and simplify to convert the equation into slope-intercept form: \[ y - 8 = -0.2x - 1 \]Add 8 to both sides to simplify: \[ y = -0.2x + 7 \]

Step 5: Write the Final Equation

The equation of the line, in slope-intercept form, is \( y = -0.2x + 7 \).

Key Concepts

Parallel Lines

Parallel lines are two lines in the same plane that never intersect. Understanding parallel lines is key for geometry and algebra because they have identical slopes. When identifying parallel lines, remember:

• They always run equidistant and in the same direction.
• If two lines have the same slope, they are parallel.

Knowing the concept of slope is essential here. The slope, represented by the letter \( m \), indicates the steepness or incline of a line. Two lines will be parallel if and only if their slopes are equal. For instance, in the equation \( y = -0.2x + 6 \), the slope is \( -0.2 \). Any line that wishes to remain parallel to this must also have a slope of \( -0.2 \). Recognizing parallel lines is thus a straightforward matter of comparing the slopes.

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know the slope of the line and one point on the line. This form is given by:

• \( y - y_1 = m(x - x_1) \) where \((x_1, y_1)\) is the known point and \(m\) is the slope.

This method is particularly advantageous when you have a point, say \((-5, 8)\), and know the slope, \( -0.2 \), as in our exercise about parallel lines. You can substitute these values directly into the formula:

• \( y - 8 = -0.2(x + 5) \)

This balances the equation around a given point, making it a quick technique to transition into slope-intercept form, immediately leading us to equation manipulation for further simplifications.

Equation of a Line

Equations of a line provide a clear mathematical expression of a straight line. There are different forms of line equations, but one commonly used in algebra is the slope-intercept form, written as:

• \( y = mx + c \)

In this format, \( m \) is the slope and \( c \) is the y-intercept, which indicates where the line crosses the y-axis.
To convert from point-slope form into slope-intercept form, distribute and simplify the equation. From:

• \( y - 8 = -0.2(x + 5) \)
• Simplify: \( y - 8 = -0.2x - 1 \)
• Finally, rearrange to: \( y = -0.2x + 7 \)

This adjustment completes the transformation into the much simpler slope-intercept form, which provides a quick reference to the line's slope and y-intercept, vital for graphing or further calculations.