Problem 24 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c}x & y \\\\\hline-2.4 & 5.2 \\\1.3 & -24.4 \\\1.75 & -28 \\\2.98 & -37.84\end{array}$$

Step-by-Step Solution

Answer

The equation of the line is \( y = -8x - 14 \).

Step 1: Understand the Slope-Intercept Form

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

Step 2: Choose Two Points from the Data

Select any two points from the provided data table. For this case, let's choose the points \((-2.4, 5.2)\) and \( (1.3, -24.4) \).

Step 3: Calculate the Slope (m)

To find the slope \( m \), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the coordinates:\[ m = \frac{-24.4 - 5.2}{1.3 - (-2.4)} = \frac{-24.4 - 5.2}{1.3 + 2.4} = \frac{-29.6}{3.7} = -8 \]

Step 4: Use Point-Slope Form to Find y-intercept (b)

With the calculated slope \( m = -8 \) and using one of the points, say \((-2.4, 5.2)\), substitute into the point-slope form equation \( y = mx + b \) to solve for \( b \):\[ 5.2 = -8(-2.4) + b \]\[ 5.2 = 19.2 + b \]\[ b = 5.2 - 19.2 = -14 \]

Step 5: Write the Equation in Slope-Intercept Form

Now that we have the slope \( m = -8 \) and the y-intercept \( b = -14 \), the equation of the line in slope-intercept form is:\[ y = -8x - 14 \]

Key Concepts

Linear Equations

A linear equation creates a straight line when graphed on a coordinate plane. The fundamental form of a linear equation is the slope-intercept form, represented by the equation \( y = mx + b \). Here:\

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• \( y \) is the dependent variable you want to solve for or describe
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• \( x \) is the independent variable or input
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• \( m \) symbolizes the slope of the line
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• \( b \) signifies the y-intercept, the point where the line crosses the y-axis
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\This equation helps depict how one quantity changes in relation to another, which is especially useful in real-world scenarios like predicting growth or trends.

Slope Calculation

Calculating the slope \( m \) of a line is a key step in forming a linear equation. Think of the slope as a measure of how steep a line is. The formula to find the slope when given two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula tells us:

• If the slope \( m \) is positive, the line rises as it moves from left to right.
• A negative slope indicates the line falls as it goes from left to right.
• A slope of zero means the line is perfectly horizontal.
• An undefined slope suggests a vertical line.

Using this technique, the slope gives insight into the rate of change between the two variables in question.

Y-Intercept

The y-intercept \( b \) is a fundamental part of the slope-intercept form, \( y = mx + b \). It is the value of \( y \) when \( x = 0 \), indicating where the line crosses the y-axis. To find the y-intercept when you have the slope \( m \) and a point \((x, y)\) on the line, plug the values into the linear equation to solve for \( b \):

• Start by substituting the \( x \) and \( y \) values into the equation \( y = mx + b \).
• Rearrange to isolate \( b \).

For example, with a slope of \( m = -8 \) and a point \((-2.4, 5.2)\), substituting these into the equation allows you to find that \( b = -14 \). Understanding the y-intercept helps visualize where a line begins or intersects the axis, crucial for graphing and interpreting the equation.