Problem 18 - 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. Through \((-5,4)\) and \((-3,2)\)

Step-by-Step Solution

Answer

The equation is \(y = -x - 1\).

Step 1: Identify the general formula for slope

We need to find the slope-intercept form of the line, which is of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. First, we determine the slope, \(m\), using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (-5, 4)\) and \((x_2, y_2) = (-3, 2)\).

Step 2: Substitute the coordinates into the slope formula

Substitute the coordinates of the points \((-5, 4)\) and \((-3, 2)\) into the slope formula:\[ m = \frac{2 - 4}{-3 - (-5)} = \frac{-2}{-3 + 5} = \frac{-2}{2} = -1 \] Thus, the slope \(m\) is \(-1\).

Step 3: Substitute into the slope-intercept form

Now that we have the slope \(m = -1\), we use one of the points to find the y-intercept \(b\). We'll use the point \((-5, 4)\):Start with the equation \(y = mx + b\) and substitute \(x = -5\), \(y = 4\), and \(m = -1\):\[ 4 = (-1)(-5) + b \]

Step 4: Solve for the y-intercept

Simplify the equation:\[ 4 = 5 + b \]Subtract 5 from both sides to solve for \(b\):\[ 4 - 5 = b \]\[ b = -1 \]

Step 5: Write the slope-intercept form of the line

Now we have the slope \(m = -1\) and the y-intercept \(b = -1\). Substitute these into the slope-intercept form equation \(y = mx + b\):\[ y = -1x - 1 \]This simplifies to:\[ y = -x - 1 \]

Key Concepts

Slope Calculation

Finding the slope of a line is essential when working with linear equations. The slope indicates how steep a line is, or how much it rises or falls as you move along the line. In mathematical terms, the slope is the change in the vertical direction divided by the change in the horizontal direction. A common way to express this is by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \(m\) represents the slope.
• \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.

To find the slope between two given points, you subtract the y-coordinates (the difference in height) and divide by the subtraction of the x-coordinates (the difference in horizontal position). For example, using points (-5, 4) and (-3, 2), the calculation would be:\[ m = \frac{2 - 4}{-3 - (-5)} = \frac{-2}{2} = -1 \]This means the slope of the line is -1. A negative result indicates that the line descends as it moves from left to right.

Equation of a Line

The equation of a line helps you describe all the points that lie on that line in a coordinate plane. One of the simplest and most commonly used forms of a line's equation is the slope-intercept form.The slope-intercept form of a line is expressed as:\[ y = mx + b \]In this form:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.
• \( y \) is the dependent variable and \( x \) is the independent variable.

To construct the equation of a line, you need both the slope and one point on the line. After calculating the slope, you can substitute it and the coordinates of the specific point into the equation to solve for the y-intercept, \(b\). This method lets you quickly express a line in terms of its slope and its point of intersection with the y-axis.

Y-Intercept

The y-intercept in the equation of a line is the value of \(y\) when \(x\) is zero. It is a crucial part of the slope-intercept form because it determines where the line crosses the y-axis. Understanding the y-intercept can help you easily draw the line on a graph since it's the starting point when \(x = 0\).In the slope-intercept equation \( y = mx + b \), the \(b\) represents the y-intercept. To find \(b\), once you have the slope \(m\) and a point \((x, y)\) on the line, substitute these into the equation:\[ y = mx + b \]Taking a point from our example, say (-5, 4) with slope -1, the equation becomes:\[ 4 = (-1)(-5) + b \]Simplifying this:

• First, calculate \(-1\times-5 = 5\).
• Then set up \(4 = 5 + b\).
• Finally, solve for \(b\) by subtracting 5 from both sides: \(b = 4 - 5 = -1\).

Thus, \(b = -1\). This means the line crosses the y-axis at the point (0, -1), completing the equation of the line as \( y = -x - 1 \). Knowing the y-intercept makes graphing lines significantly easier.