Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c} x & y \\ -7 & -44 \\ -6 & -36 \\ -5 & -28 \\ -4 & -20 \end{array}$$

Step-by-Step Solution

Answer

The slope-intercept form is \( y = 8x + 12 \).

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.

Step 2: Calculating the Slope

To find the slope \( m \), we take two points from the table. Let's use points \((-7, -44)\) and \((-6, -36)\). The formula for the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute in the values: \( m = \frac{-36 - (-44)}{-6 - (-7)} = \frac{8}{1} = 8 \). Thus, the slope \( m = 8 \).

Step 3: Using the Slope to Find the Y-Intercept

We know \( m = 8 \) from Step 2. Now substitute any point from the table into the slope-intercept form \( y = mx + b \) to find \( b \). Using point \((-7, -44)\): \( -44 = 8(-7) + b \). This simplifies to \( -44 = -56 + b \), so \( b = 12 \).

Step 4: Writing the Equation

With \( m = 8 \) and \( b = 12 \), substitute these values into the slope-intercept form \( y = mx + b \). Therefore, the equation of the line is \( y = 8x + 12 \).

Key Concepts

Linear Equations

A linear equation is like a straight path on a graph. It is a type of mathematical equation that describes a straight line. In the world of math, it's represented by the formula: \( y = mx + b \). Here, \( y \) is the dependent variable, usually found on the vertical axis. Meanwhile, \( x \), the independent variable, appears on the horizontal axis. The equation shows how \( y \) changes as \( x \) changes. Linear equations are common because they can be used to describe simple relationships between variables. They help us understand how quantities change in relation to each other. When you see a table of values, like the one from the exercise, you are essentially looking at points that belong to a linear equation. If you plot these points on a graph, they should fall on a straight line.

Slope Calculation

To find the slope of a line from two points, we can use a simple calculation. The slope measures how steep the line is. It's like how slanted or flat the line appears when plotted on a coordinate plane. The formula to calculate the slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this formula, \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two different points on the line. Let's take the points

• \((-7, -44)\) and \((-6, -36)\)

and plug them into the formula. Think of it as finding how much the \( y \) values increase or decrease - that is, the change in \( y \) - for a one-unit change in \( x \). Solve:

• \( m = \frac{-36 - (-44)}{-6 - (-7)} = \frac{8}{1} = 8 \)

This slope of 8 tells us that for each increase of 1 in \( x \), \( y \) increases by 8. It's a consistent rate of change on our line!

Y-Intercept

The y-intercept is a very special part of a linear equation. It gives us the exact spot where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). It tells us the value of \( y \) when \( x = 0 \). To find the y-intercept, we can use the slope and any point from our table. We already know our slope \( m = 8 \). Let's use the point \((-7, -44)\): First, substitute this point into the slope-intercept form:

• \(-44 = 8(-7) + b \)

Solve for \( b \):

• \(-44 = -56 + b \)
• \(b = 12 \)

This tells us that our line crosses the y-axis at \( y = 12 \). Now we have all we need to write our equation: \( y = 8x + 12 \). The y-intercept was the missing piece needed to put it all together!