Problem 81 - Precalculus

Question

The line \(y=m x+b\) connects the points (-2,4) and \((4,-2) .\) Find the values of \(m\) and \(b\).

Step-by-Step Solution

Answer

The slope is \( m = -1 \) and the y-intercept is \( b = 2 \).

Step 1: Understand the Slope Formula

The slope of a line, represented by \( m \), is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This essentially gives the 'rise' over 'run' between two points on the line.

Step 2: Substitute the Given Points into the Slope Formula

Using the points \((-2,4)\) and \((4,-2)\), substitute into the slope formula: \[ m = \frac{-2 - 4}{4 - (-2)} \].

Step 3: Calculate the Slope

Simplify the expression from Step 2: \[ m = \frac{-2 - 4}{4 + 2} = \frac{-6}{6} = -1 \]. Therefore, the slope \( m = -1 \).

Step 4: Use the Point-Slope Form to Find the Intercept

Now use one of the points, for instance \((4,-2)\), in the point-slope form of a line equation: \[ y - y_1 = m(x - x_1) \]. This can be rewritten as: \[ y + 2 = -1(x - 4) \].

Step 5: Simplify to Get the Equation

Distribute and simplify the equation from Step 4: \[ y + 2 = -x + 4 \].

Step 6: Solve for the y-intercept

Rearrange the equation to solve for \( y \): \[ y = -x + 4 - 2 \], which simplifies to \( y = -x + 2 \). This shows that the y-intercept \( b = 2 \).

Key Concepts

Slope

The slope of a line is a key concept in understanding linear relationships. It is denoted by the letter \( m \) and tells us how steep a line is and in which direction it slopes. The formula for calculating the slope when you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

This formula calculates the 'rise' (change in \( y \)) over the 'run' (change in \( x \)) between the two points.
For instance, between the points \((-2, 4)\) and \((4, -2)\), you substitute these values into the formula:

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

The negative slope indicates the line decreases as it moves from left to right on a graph.
Understanding the concept of slope helps in predicting and analyzing tendencies in data, such as speed or rate of change.

Y-Intercept

The y-intercept of a line, represented by \( b \), is the point where the line crosses the y-axis. This is important because it tells us the starting point of the line when \( x = 0 \).
In the slope-intercept form of a line equation, \( y = mx + b \), \( b \) is the y-intercept. To find this value, once the slope \( m \) is known, simply use any point on the line to solve for \( b \).
Using our previous example, after finding the slope \( m = -1 \), you can use one of the given points, such as \((4, -2)\), in the equation:

• \( y = -x + b \)

Plug in \( y = -2 \) and \( x = 4 \), then solve for \( b \):

• \(-2 = -1(4) + b\)

This simplifies to:\( b = 2 \). The y-intercept \( b = 2 \) is where the line crosses the y-axis.

Point-Slope Form

The point-slope form of a linear equation is a powerful tool when you want to write the equation of a line given a point and the slope. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

This form showcases the slope \( m \) and a specific point on the line \( (x_1, y_1) \).
It's particularly useful because it allows easy construction of the line's equation without first converting to slope-intercept form. For our points \((4, -2)\) and with slope \( m = -1 \), the formula becomes:

• \( y + 2 = -1(x - 4) \)

After rearranging and simplifying, you arrive at the slope-intercept form \( y = -x + 2 \). This conversion demonstrates how point-slope and slope-intercept forms are connected.
Additionally, point-slope form provides insight into the relationship between a single point on the line and the general direction in which the line is moving.