Problem 26 - 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. $$\begin{array}{c|c}x & y \\\\\hline-1.1 & 1.5 \\\\-1.0 & 2.0 \\\\-0.9 & 2.5 \\\\-0.8 & 3.0\end{array}$$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = 5x + 7.0\).

Step 1: Understanding the Point Coordinates

The table gives us four points through which the line passes: \((-1.1, 1.5), (-1.0, 2.0), (-0.9, 2.5), (-0.8, 3.0)\). These points suggest that the line has a consistent pattern, implying it is a linear relation. We aim to find the slope \(m\) and the y-intercept \(b\) for the line's equation \(y = mx + b\).

Step 2: Calculating the Slope

The slope \(m\) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[m = \frac{y_2 - y_1}{x_2 - x_1}.\] Choosing two consecutive points: For \((-1.0, 2.0)\) and \((-0.9, 2.5)\) \[m = \frac{2.5 - 2.0}{-0.9 - (-1.0)} = \frac{0.5}{0.1} = 5.\] Thus, the slope \(m\) is 5.

Step 3: Finding the Y-Intercept

Substitute one point and the slope \(m = 5\) into the equation \(y = mx + b\) to solve for \(b\). Using \((-1.0, 2.0)\): \[2.0 = 5(-1.0) + b\] \[2.0 = -5.0 + b\] \[b = 7.0.\] Hence, the y-intercept \(b\) is 7.0.

Step 4: Writing the Equation in Slope-Intercept Form

Now that we have \(m = 5\) and \(b = 7.0\), the slope-intercept form of the equation is \[y = 5x + 7.0.\] This equation describes the line passing through the given points with a consistent increase in \(y\) by 5 for every 0.1 increase in \(x\).

Key Concepts

Slope Calculation

To find the slope of a line, you need to determine how steep the line is. The slope is represented by the letter \(m\) in the equation of a line, and it defines how much the \(y\)-value changes for a change in the \(x\)-value. Imagine you're hiking up a hill; the slope tells you how steep your path is.
In mathematical terms, the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

When you choose any two points from a table of points that represent a line, like \((-1.0, 2.0)\) and \((-0.9, 2.5)\), you can compute the slope. Here, the difference in \(y\) is \(2.5 - 2.0 = 0.5\) and the difference in \(x\) is \(-0.9 - (-1.0) = 0.1\).
Dividing these differences gives us:

• \(m = \frac{0.5}{0.1} = 5\)

This means, for every unit increase along the \(x\)-axis, the \(y\)-value increases by 5. Calculating slope is an essential step in identifying the relationship represented by a linear equation.

Linear Equations

A linear equation creates a straight line when graphed on a coordinate plane. The simplest form of a linear equation is the slope-intercept form, written as \(y = mx + b\). This format helps you to quickly understand the rate of change and where the line crosses the \(y\)-axis.
Here's a breakdown:

• \(m\) represents the slope of the line.
• \(b\) is the \(y\)-intercept, which we'll discuss further in the next section.

In our example, after calculating the slope \(m = 5\), we use known point values to find \(b\). This ensures we have all components to create a full equation that represents any point along the line.
Linear equations are powerful for predicting and understanding relationships between two variables. When you graph them, they appear as straight lines, which indicates a constant rate of change.

Y-Intercept

The \(y\)-intercept is the point where the line crosses the \(y\)-axis on a graph. In the slope-intercept form \(y = mx + b\), \(b\) gives us this intersection point. It is the value of \(y\) when \(x\) is zero.
To find the \(y\)-intercept from the exercise, we use the slope calculated and any given point. By substituting the slope and one point, such as \((-1.0, 2.0)\), into the equation, we solve for \(b\):

• \(2.0 = 5(-1.0) + b\)
• \(2.0 = -5.0 + b\)
• \(b = 7.0\)

Thus, the \(y\)-intercept \(b\) is 7.0, meaning the line hits the \(y\)-axis at \(y = 7.0\). This value is crucial in graphing as it gives us a starting point from where the line begins its journey across the graph. Understanding the \(y\)-intercept provides insight into where a trend or relationship starts.