Problem 25 - 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. $$\begin{array}{c|c} x & y \\ \hline 2 & -5 \\ 3 & -8 \\ 4 & -11 \\ 5 & -14 \end{array}$$

Step-by-Step Solution

Answer

The slope-intercept form of the line is \(y = -3x + 1\).

Step 1: Identify Two Points

From the given table, select any two points. Let's choose the points \((2, -5)\) and \((3, -8)\).

Step 2: Calculate the Slope

Use the formula for the slope \(m\) which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points \((2, -5)\) and \((3, -8)\), we get:\[m = \frac{-8 - (-5)}{3 - 2} = \frac{-8 + 5}{1} = \frac{-3}{1} = -3\.\]

Step 3: Use Point-Slope Form

With the slope \(m = -3\) and the point \((2, -5)\), plug these into the point-slope form of the equation, \(y - y_1 = m(x - x_1)\):\[y - (-5) = -3(x - 2)\.\]

Step 4: Simplify to Slope-Intercept Form

Expand the equation to get the slope-intercept form \(y = mx + b\):\[y + 5 = -3x + 6\quad \Rightarrow \quad y = -3x + 6 - 5 \quad \Rightarrow \quad y = -3x + 1\.\]

Step 5: Verify the Equation

Substitute the other points \((4, -11)\) and \((5, -14)\) into the equation \(y = -3x + 1\) to ensure consistency:For \((4, -11)\): \(-11 = -3(4) + 1 = -12 + 1 = -11\), correct.For \((5, -14)\): \(-14 = -3(5) + 1 = -15 + 1 = -14\), correct.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is incredibly useful when you need to write the equation of a line, knowing only its slope and a single point on the line. This form is represented as:

• \( y - y_1 = m(x - x_1) \)

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. This form is perfect for when you don't know the y-intercept yet.
To use the point-slope form, simply substitute the slope (previously calculated) and any point on the line into the formula. For example, if the slope is \(-3\) and our point is \((2, -5)\), as shown in the step-by-step solution, it looks like this:

• \( y - (-5) = -3(x - 2) \)

With this setup, you can easily rearrange the equation into the standard or slope-intercept form if needed.

Linear Equations

Linear equations are equations of the first degree, meaning the variables are in the first power and graphically they represent straight lines. The general form of a linear equation in two dimensions is:

• \(ax + by = c\)

A more common representation, especially in graphing, is the slope-intercept form \(y = mx + b\). Here, \(m\) is the slope indicating the line's steepness and direction, while \(b\) provides the y-coordinate of the point where the line intersects the y-axis.
Linear equations are important in representing consistent rates of change. Each increase or decrease in the x-variable leads to a change in the y-variable, dictated by the slope. This consistency makes them quite predictable and useful in modeling real-world phenomena like speed, economics, etc.
Understanding how these equations work allows you to easily convert among different forms: general, standard, point-slope, and slope-intercept, ensuring you have the right tool for any problem at hand.

Calculating Slope

At the heart of linear equations is the concept of slope. The slope is a measure of how steep a line is. It's calculated by determining the ratio of the 'rise' over the 'run' between two points on the line. In mathematical terms:

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

The "rise" refers to the difference in the y-values, and the "run" refers to the difference in the x-values. For example, using points \((2, -5)\) and \((3, -8)\), the slope \(m\) is:

• \( m = \frac{-8 + 5}{3 - 2} = -3 \)

A positive slope means the line goes upwards as you move from left to right, while a negative slope means it goes downwards. A zero slope results in a horizontal line, and an undefined slope results in a vertical line. Understanding these characteristics helps you quickly analyze and understand the line's behavior. Calculating the slope correctly is essential for constructing and interpreting linear equations accurately.