Problem 22 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((-1,6)\) and \((2,-3)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Points

First, identify the two given points on the line: - Point 1: \((-1, 6)\)- Point 2: \((2, -3)\)

Step 2: Calculate the Slope

To find the slope \(m\), use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates of the points:\[ m = \frac{-3 - 6}{2 - (-1)} = \frac{-9}{3} = -3 \]

Step 3: Use the Point-Slope Form

Use the point-slope form of a line equation:\[ y - y_1 = m(x - x_1) \]Choose the point \((-1, 6)\):\[ y - 6 = -3(x + 1) \]

Step 4: Expand and Simplify

Expand the equation from the point-slope form:\[ y - 6 = -3x - 3 \]Simplify to find the slope-intercept form \(y = mx + b\):\[ y = -3x + 3 \]

Step 5: Verify the Equation

Verify by checking the other point \((2, -3)\):Substitute \(x = 2\) into \(y = -3x + 3\):\[ y = -3(2) + 3 = -6 + 3 = -3 \]Since both points satisfy the equation, it confirms the solution is correct.

Key Concepts

Linear Equations

Linear equations form the backbone of algebra, often expressed in the form of \(ax + by = c\). For a linear equation in two dimensions, this relationship represents a straight line when graphed. The simplest form of a linear equation is the slope-intercept form, \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. In this exercise, we used this form to express a line that passes through two specific points. It's crucial to understand that a linear equation establishes a constant rate of change between the variables. This is why the graph is always a straight line.
Here’s why it’s useful:

• Simplicity: It's easy to identify the slope and y-intercept directly.
• Graphing: Easily graph a line using the slope and y-intercept.
• Comparison: Quickly compare different linear equations to understand how lines relate or cross.

Linear equations appear in various real-world applications, such as calculating speed given distance and time or converting temperatures between Fahrenheit and Celsius.

Slope Calculation

The slope of a line is a measure of its steepness and direction. It is crucial when discussing linear equations because it dictates how the line moves across the graph. The formula for slope, denoted by \(m\), is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here’s a deeper dive into what this formula means:
- **Change in y (abla y)**: Represents how much the y-value changes between two points. In our example, this was \(-3 - 6 = -9\).
- **Change in x (abla x)**: Represents the change of the x-value, calculated as \(2 - (-1) = 3\).
- **Ratio**: The slope is the ratio of these changes: \(\frac{-9}{3} = -3\). This negative slope tells us that as we move from left to right, the line decreases, meaning it goes downhill. Slope calculation is essential for understanding the relationship between variables on a graph and determining whether the line rises or falls.

Point-Slope Form

Sometimes, we know a point on the line and its slope but not its equation. The point-slope form helps in these scenarios. It is expressed as \(y - y_1 = m(x - x_1)\) where \( (x_1, y_1) \) is a specific point on the line, and \(m\) is the slope. Here’s a quick outline of how it works:

• Select a point: We chose \((-1, 6)\) as our known point.
• Substitute: We input \(m = -3\) and \((x_1, y_1)\) into the formula, yielding \(y - 6 = -3(x + 1)\).
• Expand and Simplify: Expanding results in \(y - 6 = -3x - 3\), which simplifies to \(y = -3x + 3\).

This form becomes foundational when deriving the equation of a line from known data points for graphing or analysis. It's an intermediate step, making calculation straightforward before arriving at the slope-intercept form.