Problem 18 - Thomas Calculus

Question

In Exercises 17–30, write an equation for each line described. Passes through \((2,-3)\) with slope 1\(/ 2\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{1}{2}x - 4 \).

Step 1: Identify the Slope-Intercept Form

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

Step 2: Substitute the Given Slope

We know that the slope \( m = \frac{1}{2} \). Substitute this value into the slope-intercept form equation: \( y = \frac{1}{2}x + b \).

Step 3: Use the Point to Find the Y-Intercept

To find \( b \), substitute the coordinates of the given point \((2, -3)\) into the equation. This gives \( -3 = \frac{1}{2}(2) + b \).

Step 4: Solve for the Y-Intercept

Simplify the equation to solve for \( b \): \( -3 = 1 + b \). Subtract 1 from both sides to get \( b = -4 \).

Step 5: Write the Final Equation

Now that you have both the slope and the y-intercept, write the final equation of the line: \( y = \frac{1}{2}x - 4 \).

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that describes all the points along a particular line in a coordinate plane. The most common forms of linear equations are:

• The Slope-Intercept Form: \( y = mx + b \)
• The Point-Slope Form: \( y - y_1 = m(x - x_1) \)

Each of these forms serves a unique purpose and provides valuable information about the line, such as its slope and points through which it passes.
Being familiar with these forms is essential for identifying the characteristics of a line and graphing it accurately.

Point-Slope Formula

The Point-Slope formula is a useful tool when you know a point on a line and its slope. It's expressed as \( y - y_1 = m(x - x_1) \), where

• \( m \) is the slope of the line,
• \( (x_1, y_1) \) is the given point on the line.

This formula is particularly helpful when you need to quickly write an equation for a line. By simply plugging in the slope and the coordinates, you can find the equation with ease.
Unlike the slope-intercept form, you can derive the full equation without knowing the y-intercept ahead of time, allowing for more flexibility in problem-solving.

Y-Intercept

The y-intercept is where a line crosses the y-axis in the coordinate plane. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).
Having this information allows you to understand where the line starts and can be crucial for graphing. For example, to find the y-intercept from the problem statement, we substitute the point \((2, -3)\) into the equation \( y = \frac{1}{2}x + b \) and solve for \( b \).
Once we solve, we determine that \( b = -4 \), indicating the line crosses the y-axis at -4. This position helps in aligning other points accurately on the graph.

Slope Calculation

Calculating the slope is one of the first and most essential steps in understanding the behavior of a line. Slope is defined as \( m = \frac{\text{rise}}{\text{run}} \), which translates to the change in y over the change in x.
The slope tells you how steep a line is and the direction it moves. In the problem we solve, the slope is given as \( \frac{1}{2} \), meaning the line rises by 1 unit for every 2 units it moves to the right.

• A positive slope indicates a line that rises upward.
• A negative slope shows a downward decline.
• A zero slope means the line is horizontal, while an undefined slope shows a vertical line.

This understanding is crucial for predicting how a line will move across the Cartesian plane and relates directly to how real-world data is often interpreted.