Problem 7 - College Algebra with Modeling and Visualization

Question

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Passing through \((1,-2)\) and \((-9,3)\)

Step-by-Step Solution

Answer

The slope-intercept form of the line is \(y = -\frac{1}{2}x - \frac{3}{2}\) .

Step 1: Calculate the Slope

To find the slope (\(m\)) of the line passing through the points \((1, -2)\) and \((-9, 3)\), use the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points: \(m = \frac{3 - (-2)}{-9 - 1} = \frac{5}{-10} = -\frac{1}{2}\). Thus, the slope is \(-\frac{1}{2}\).

Step 2: Write the Point-Slope Form

With the slope \(m = -\frac{1}{2}\) and the point \((1, -2)\), we use the point-slope form equation: \(y - y_1 = m(x - x_1)\). Substitute the values: \(y - (-2) = -\frac{1}{2}(x - 1)\), which simplifies to \(y + 2 = -\frac{1}{2}(x - 1)\).

Step 3: Convert to Slope-Intercept Form

Start with the point-slope equation \(y + 2 = -\frac{1}{2}(x - 1)\) and distribute the slope: \(y + 2 = -\frac{1}{2}x + \frac{1}{2}\). Then, isolate \(y\) by subtracting 2 from both sides: \(y = -\frac{1}{2}x + \frac{1}{2} - 2\). Simplify to get \(y = -\frac{1}{2}x - \frac{3}{2}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest and most commonly used forms to write the equation of a line. It is expressed as \(y = mx + c\), where:

• \(m\) is the slope of the line, which indicates how steep the line is.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is especially useful because it allows you to quickly identify both the slope and the y-intercept of a line.
In our example, after simplifying, the slope-intercept form of the line passing through points \((1, -2)\) and \((-9, 3)\) is \(y = -\frac{1}{2}x - \frac{3}{2}\). Here, the slope \(m\) is \(-\frac{1}{2}\), and the y-intercept \(c\) is \(-\frac{3}{2}\). This tells us that the line is decreasing, meaning it goes downward as you move from left to right across the graph.
Understanding the slope-intercept form makes graphing lines and solving algebraic problems involving lines very intuitive.

Equation of a Line

The equation of a line is a mathematical expression that gives us all the points lying on that line.
Multiple forms can be used to represent a line such as the point-slope form, standard form, and the slope-intercept form. Each of these can be useful in different scenarios.

• The **point-slope form** is \(y - y_1 = m(x - x_1)\), beneficial when you know one point and the slope.
• The **standard form** is \(Ax + By = C\), often used for integer coefficients and analysis involving whole numbers.
• The **slope-intercept form** discussed above is \(y = mx + c\).

In our exercise, we started with the point-slope form using the point \((1, -2)\) and slope \(m = -\frac{1}{2}\).
We then converted it to find the slope-intercept form. Each form addresses different needs, but they all derive from the fundamental equation of a line, which is about finding the relationship between \(x\) and \(y\). Mastering conversions between these forms helps in visualizing and solving geometric and algebra problems effectively.

Slope Calculation

The slope of a line is a measure of its steepness and direction. It is calculated using the change in \(y\) over the change in \(x\) between two distinct points.
Mathematically, the formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This ratio tells us how much \(y\) changes for a unit change in \(x\).

• If the slope \(m\) is positive, the line of interest is climbing upwards as you move from left to right.
• If \(m\) is negative, the line descends.
• A zero slope means the line is horizontal, indicating a constant function with no rise or fall.
• An undefined slope (where the denominator is zero) indicates a vertical line.

In the given exercise, two points \((1, -2)\) and \((-9, 3)\) were used. Plugging them into the formula, we get \(m = \frac{3 - (-2)}{-9 - 1} = -\frac{1}{2}\), illustrating a line that decreases by half a unit for every unit increase in \(x\).
Understanding slopes is critical as it allows prediction and analysis of the line's behavior without graphing it.