Problem 21 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((-1,-4)\) and \((1,2)\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = 3x - 1 \).

Step 1: Identify the Formula for Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \( m \) is the slope.

Step 2: Substitute the Points into the Formula

Use the points \((-1, -4)\) and \((1, 2)\). Substitute into the slope formula:\[ m = \frac{2 - (-4)}{1 - (-1)} = \frac{6}{2} = 3 \]Thus, the slope \( m = 3 \).

Step 3: Use Point-Slope Form to Find Equation

The point-slope form of a line is \[ y - y_1 = m(x - x_1) \]Using point \((-1, -4)\) and slope \( m = 3 \), substitute into the formula:\[ y + 4 = 3(x + 1) \]

Step 4: Simplify to Slope-Intercept Form

Distribute and simplify the equation:\[ y + 4 = 3x + 3 \]Subtract 4 from both sides to solve for \( y \):\[ y = 3x - 1 \]

Step 5: Finalize the Slope-Intercept Form

Now, the equation is in slope-intercept form: \[ y = mx + b \], where \( m = 3 \) and \( b = -1 \).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a powerful tool for quickly constructing the equation of a line when you know a point on the line and its slope. The formula looks like this:

• \( y - y_1 = m(x - x_1) \)

In this equation,

• \((x_1, y_1)\) represents a known point through which the line passes,
• \(m\) is the slope,
• \(x\) and \(y\) are the variables that express the coordinates of any generic point on the line.

Let's imagine that you have a point

• for example, \((-1, -4)\),

and a slope

• say, \(m = 3\).

Plug these into the point-slope formula to reveal the equation of the line.

• \( y + 4 = 3(x + 1) \)

This form is easy to use and provides a direct way to confirm linear equations from given conditions. Once you have this form, it's straightforward to rearrange it into any other preferred format, like the slope-intercept form, which is often more intuitive to interpret.
Use the point-slope form when you have a point and a slope at your disposal and need to derive the full linear equation.

Slope Calculation

Calculating the slope of a line is a key step in understanding its direction and steepness. To find the slope, you need two points on the line. The formula for calculating slope is:

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

Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points,
• \(m\) represents the slope.

This formula demonstrates the change in \(y\) (vertical change) relative to the change in \(x\) (horizontal change).

• Essentially, it tells you how much \(y\) increases or decreases for every unit \(x\) moves.

If we take the points from our example,

• \((-1, -4)\) and \((1, 2)\),
• substitute them into the slope formula,

we calculate the slope \(m\) as follows:

• \( m = \frac{2 - (-4)}{1 - (-1)} = \frac{6}{2} = 3 \).

Understanding slope makes recognizing a line's lean more intuitive, and it simplifies comparing the inclinations of different lines. Whether dealing with horizontal lines \((m = 0)\), vertical lines \((m\) is undefined), or any sloping line, this calculation is fundamental.

Linear Equations

Linear equations are fundamental in algebra and describe straight lines. They are represented in several forms, such as the slope-intercept form \( y = mx + b \), where:

• \(m\) represents the slope of the line,
• \(b\) is the y-intercept, where the line crosses the y-axis.

Linear equations are characterized by their constant rate of change, which means the graph of a linear equation is always a straight line.
Let's consider turning the equation from point-slope form to slope-intercept form. For the given exercise, starting with the equation:

• \( y + 4 = 3(x + 1) \)

We can simplify it to express \(y\) explicitly:

• After distributing and rearranging, we have \( y = 3x - 1 \).

This is the slope-intercept form, clearly showing that our line has a slope of 3 and crosses the y-axis at \(-1\).
Linear equations are not only a pivotal part of algebra but also serve critical roles in various fields, from economics where they model relationships and trends, to physics where they describe motion and forces. Understanding linear equations and their forms allows you to beautifully track and predict patterns in both mathematical and real-world contexts.