Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) $$\text { Through }(2,4), m=-1$$

Step-by-Step Solution

Answer

The equation is \( y = -x + 6 \).

Step 1: Understand the Slope-Intercept Form Equation

The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Identify the Given Slope and Point

From the exercise, we know the slope \( m = -1 \) and the point through which the line passes is \( (2, 4) \).

Step 3: Substitute the Slope and Point into the Equation

Substitute \( m = -1 \), \( x = 2 \), and \( y = 4 \) into the slope-intercept form \( y = mx + b \) to find \( b \). The equation becomes \( 4 = -1 \times 2 + b \).

Step 4: Solve for the Y-Intercept (b)

Solve the equation \( 4 = -2 + b \) : Add \( 2 \) to both sides to isolate \( b \), resulting in \( b = 4 + 2 \), which simplifies to \( b = 6 \).

Step 5: Write the Final Equation

Substitute \( m = -1 \) and \( b = 6 \) back into the slope-intercept form, giving us the final equation: \( y = -x + 6 \).

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they have the general form of ax + b = c, where a, b, and c are constants. They represent straight lines on a coordinate plane and are foundational in algebra.

In real-world applications, linear equations are everywhere. They're used in calculating areas, predicting profits and losses, and even in determining speed and distance. A key property of a linear equation is its graph, which is always a straight line. The simplicity of linear equations makes them an excellent tool for approximating complex situations.

The slope-intercept form is a popular version of a linear equation because it clearly shows both the slope and the y-intercept of the line, making it highly useful for graphing.

Slope

The slope of a line is a measure of how steep it is. It describes the direction and the incline of the line.

• If the slope is positive, the line inclines upward as it moves from left to right.
• If the slope is negative, the line declines downward as it moves from left to right.
• A zero slope means the line is horizontal, and an undefined slope means the line is vertical.

In our exercise, we see a slope of -1. This tells us that for every unit increase in x, y decreases by one unit. Graphically, this translates to a line that slants downward at a 45-degree angle.

Understanding the slope is crucial for sketching the behavior of a line and predicting changes in the dependent variable for every change in the independent variable.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It provides an initial value or a starting point when graphing a line.

In the slope-intercept equation form, y = mx + b, the y-intercept is represented by b. This value is critical because it indicates where the line touches the y-axis when the value of x is zero.

• A positive y-intercept means the line starts above the origin.
• A negative y-intercept means it starts below the origin.
• If the y-intercept is zero, the line passes through the origin point.

In our solution, the y-intercept we calculated is 6, meaning the line crosses the y-axis at the point (0,6). This value plays a significant role in determining the position of the line on the graph and aids in visualizing the linear relationship defined by the equation.