Problem 72 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. Passing through (-3,7) and (1,2)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{5}{4}x + \frac{13}{4} \).

Step 1: Understand the problem

We need to find the equation of a line through two given points (-3,7) and (1,2) and express it in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the slope

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (-3,7) and (1,2), we plug in the values: \( m = \frac{2 - 7}{1 - (-3)} = \frac{-5}{4} \). The slope is \( m = -\frac{5}{4} \).

Step 3: Use point-slope form to find y-intercept

Using the point-slope form \( y - y_1 = m(x - x_1) \), we will plug in one of the points, say (1,2), and the slope to find the y-intercept. \( y - 2 = -\frac{5}{4}(x - 1) \).

Step 4: Simplify the equation

Distribute the slope on the right side: \( y - 2 = -\frac{5}{4}x + \frac{5}{4} \). Add 2 to both sides to solve for \( y \): \( y = -\frac{5}{4}x + \frac{5}{4} + 2 \).

Step 5: Find the y-intercept

Simplify \( \frac{5}{4} + 2 \) to find the y-intercept. First, convert 2 to fractional form: \( 2 = \frac{8}{4} \). Therefore, \( \frac{5}{4} + \frac{8}{4} = \frac{13}{4} \). The y-intercept \( b \) is \( \frac{13}{4} \).

Step 6: Write the final equation in slope-intercept form

Combine the slope and y-intercept values into the slope-intercept form equation: \( y = -\frac{5}{4}x + \frac{13}{4} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line. This form makes it easy to see the slope and the y-intercept of the line directly from the equation.
The general formula for this form is \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is the value of \( y \) where the line crosses the y-axis.

This form is useful because it allows quick identification of the slope and the starting point of the line on the graph.
It simplifies graphing since you can start from the y-intercept \( b \) and use the slope \( m \) to find other points on the line.

Slope Calculation

The slope of a line represents its steepness and direction. It's a measure of the rate of change of \( y \) with respect to \( x \).
To calculate the slope \( m \) between two points \( (x_1, y_1) \) and \((x_2, y_2)\), you use the formula:

\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]

This formula gives a single number that describes how much \( y \) changes for a unit change in \( x \).
If \( m \) is positive, the line goes upwards as you move from left to right.
If \( m \) is negative, the line goes downwards.
In our example, using points (-3,7) and (1,2), the slope is calculated as:

\( m = \frac{2 - 7}{1 + 3} = -\frac{5}{4} \).
This tells us the line falls 5 units in \( y \) direction for every 4 units it moves in \( x \) direction.

Point-Slope Form

The point-slope form is another way to write the equation of a line that is useful when you have one point and the slope.
The general formula is: \( y - y_1 = m(x - x_1) \).
This gives a linear equation based on a known slope \( m \), and a specific point \( (x_1, y_1) \).

Using the point-slope form makes it simple to rearrange and solve for \( y \) to eventually convert it to the slope-intercept form.
Once you input the point and slope into the point-slope formula, you can rearrange the equation to find the y-intercept (\( b \)) by solving for \( y \).

Y-Intercept

The y-intercept is a key feature of a line's equation when in slope-intercept form. It is represented by \( b \) in the equation \( y = mx + b \).
This intercept is crucial because it shows where the line crosses the y-axis, revealing the line's starting point when \( x = 0 \).
To determine \( b \), substitute a point's \( x \) and \( y \) values into the slope-intercept equation, then solve for \( b \).
In our exercise, once the slope was found as \(-\frac{5}{4}\) and using the point-slope form with point (1,2), solving the equation yielded:

\( b = \frac{13}{4} \).
This indicates that when \( x = 0 \), \( y = \frac{13}{4} \), which is the coordinate where the line meets the y-axis.