Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) $$\text { Through }(1,3), m=-2$$

Step-by-Step Solution

Answer

The equation of the line is \( y = -2x + 5 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this problem, we are given the slope \( m = -2 \).

Step 2: Substitute the Point into the Formula

Our task is to find the value of \( b \), the y-intercept. We substitute the given point \((1,3)\) into the equation \( y = mx + b \). So, substituting \( x = 1 \), \( y = 3 \), and \( m = -2 \) gives us the equation: \[ 3 = -2(1) + b \]

Step 3: Solve for the Y-Intercept

Solve the equation from Step 2: \[ 3 = -2 \times 1 + b \]\[ 3 = -2 + b \]Add \( 2 \) to both sides to solve for \( b \): \[ 3 + 2 = b \]\[ b = 5 \]

Step 4: Write the Final Equation

Substitute \( b = 5 \) back into the slope-intercept equation to get the line's equation: \[ y = -2x + 5 \]

Key Concepts

Linear Equation

A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form, which is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) denotes the y-intercept of the line. Linear equations are fundamental in algebra because they represent constant rates of change and form the basis for more complex mathematical concepts.
The slope \( m \) indicates the steepness of the line, while the y-intercept \( b \) represents the point at which the line crosses the y-axis. Together, these components allow us to understand and graph the behavior of the line effectively.

Slope and Y-Intercept

The slope and y-intercept are crucial elements in understanding linear equations.

• Slope (\( m \)): Indicates the rate of change of the line. If the slope is positive, the line ascends from left to right. A negative slope means the line descends.
• Y-intercept (\( b \)): The value of \( y \) when \( x = 0 \). It is the starting point of the line on the y-axis.

In our example, the slope is \( -2 \), which means the line will slope downward as it moves from left to right.The y-intercept, once determined, will tell us where precisely the line crosses the y-axis, establishing its position on the graph.

Solving for Y-Intercept

To solve for the y-intercept, we substitute the given point and the slope into the slope-intercept form equation. In our exercise, the given point is \((1,3)\) and the slope is \( -2 \). By substituting \( x = 1 \), \( y = 3 \), and \( m = -2 \) into the equation \( y = mx + b \), we obtain the following:
\[ 3 = -2(1) + b \] Solving this equation involves isolating \( b \):

• Calculate \( -2 \times 1 = -2 \)
• Add \( 2 \) to both sides: \( 3 + 2 = b \)
• Thus, \( b = 5 \)

The y-intercept \( b = 5 \) tells us where the line crosses the y-axis, which is crucial for graphing the equation.

Graphing Linear Equations

Graphing a linear equation involves plotting the line on a coordinate plane using the slope and y-intercept. To start, pinpoint the y-intercept on the graph. For the example equation \( y = -2x + 5 \):

• Locate \( b = 5 \) on the y-axis.
• Using the slope \( -2 \), which can be interpreted as \( -2/1 \), move down 2 units and 1 unit to the right from the y-intercept to find another point.

Repeat this slope pattern to draw several points, and then connect them to form a straight line.
Graphing visually represents the relationship expressed by the equation and helps in understanding how changes in the slope and y-intercept affect the line's position and direction on the graph.