Problem 14 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=8,\) passing through (4,-1)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 1 = 8(x - 4)\) and in slope-intercept form, it is \(y = 8x - 33\).

Step 1: Identify the Given Information

The slope of the line, denoted as m, is given as 8. A point that the line passes through is given as (4, -1). The 'x' and 'y' coordinates of this point are 4 and -1 respectively.

Step 2: Use the Point-Slope Formula

The point-slope formula is \(y - y_{1} = m(x - x_{1})\), where \(m\) is the slope, and \((x_{1}, y_{1})\are the coordinates of the given point. Insert the given values into this formula, it becomes \(y - (-1) = 8(x - 4)\). This simplifies to \(y + 1 = 8x - 32\).

Step 3: Convert to Slope-Intercept Form

The slope-intercept form is known as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. To convert the point-slope form to this form, isolate \(y\) on one side of the equation. The equation, \(y + 1 = 8x - 32\), will become \(y = 8x - 32 - 1 = 8x - 33\), which is the slope-intercept form.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a valuable tool for writing the equation of a line when you know a point on that line and its slope. This form is given by the formula:

• \( y - y_{1} = m(x - x_{1}) \)

where:

• \( m \) is the slope of the line, and
• \((x_{1}, y_{1})\) are the coordinates of the given point.

This formula is versatile because it allows us to quickly substitute the known values, such as a specific point and the slope, to find the equation.
For instance, if a line passes through the point (4, -1) and has a slope of 8, we substitute these into the formula to get:

• \( y - (-1) = 8(x - 4) \)

After simplifying, this results in the expression \( y + 1 = 8x - 32 \). Point-slope form is particularly useful for finding the equation of a line quickly and precisely without needing additional information like the y-intercept.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is very straightforward to use and interpret. The formula is:

• \( y = mx + c \)

where:

• \( m \) represents the slope,
• and \( c \) stands for the y-intercept, which is the value of \( y \) when \( x = 0 \).

To convert from point-slope form to slope-intercept form, you simply solve the equation for \( y \).
Taking the equation \( y + 1 = 8x - 32 \) from the point-slope form, you can rearrange it to slope-intercept form by isolating \( y \):

• \( y = 8x - 32 - 1 \)

This simplifies to \( y = 8x - 33 \). The benefit of the slope-intercept form is that it gives you immediate information about the slope and y-intercept, which can be very handy for graphing or understanding the line's behavior.

Linear Equations

Linear equations are fundamental in algebra and mathematics in general. These equations describe a straight line when graphed on a coordinate plane. A linear equation can be written in several forms, including point-slope and slope-intercept form. Regardless of the form, a linear equation will always depict a relationship where one variable is dependent on another in a consistent, additive manner.

The standard form of a linear equation is usually expressed as:

• \( Ax + By = C \)

where \( A, B, \) and \( C \) are constants. This form can be particularly useful for solving equations using elimination or substitution methods. However, for writing equations of lines from points and slopes, the point-slope and slope-intercept forms are often more convenient.
Key characteristics of linear equations include:

• Consistent slope \( m \) demonstrating a constant rate of change.
• A straight-line graph.

Linear equations are powerful tools in modeling real-world situations where relationships can be graphed as straight lines, such as determining the cost over time with a constant rate.