Problem 68 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. Slope \(=3,\) passes through (-3,2)

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x + 11 \).

Step 1: Understand the Formula

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.

Step 2: Identify Known Values

We are given that the slope \( m = 3 \) and the line passes through the point \( (-3, 2) \). We need to find \( b \).

Step 3: Substitute Known Values into Formula

Substitute \( m = 3 \) and the point \((-3, 2)\) into the slope-intercept equation \( y = mx + b \), which becomes \( 2 = 3(-3) + b \).

Step 4: Solve for y-intercept b

Calculate \( 3(-3) \) to get \( -9 \). Therefore, the equation becomes \( 2 = -9 + b \). Add 9 to both sides to solve for \( b \): \( b = 11 \).

Step 5: Write the Final Equation

Now that we have \( m = 3 \) and \( b = 11 \), substitute these into the slope-intercept form: \( y = 3x + 11 \). Therefore, the equation of the line is \( y = 3x + 11 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental way to express the equation of a straight line. Its formula is: \[ y = mx + b \] In this expression:

• The symbol \( y \) represents the dependent variable.
• The variable \( m \) stands for the slope of the line, which shows how steep the line is.
• The variable \( x \) is the independent variable.
• The constant \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

Using slope-intercept form allows us to quickly understand the line’s direction and where it crosses the y-axis. With these two pieces of information, we can easily graph the line or predict the value of \( y \) for any given \( x \).

Y-Intercept

The y-intercept is a crucial part of the linear equation in slope-intercept form. It is where the line crosses the y-axis, and it can be found directly from the equation as the constant term \( b \). When you set \( x = 0 \) in the equation \( y = mx + b \), the value of \( y \) you find is the y-intercept. In everyday use, the y-intercept provides a starting point on a graph. If you're plotting the equation, you know that one of your points will be immediately at \( (0, b) \). Finding the y-intercept step is straightforward:

• Substitute 0 for \( x \).
• Solve the equation to find \( b \).

For the line equation \( y = 3x + 11 \), the 11 indicates the y-intercept, meaning the line crosses the y-axis at \( (0, 11) \).

Slope Calculation

Slope describes the steepness and direction of a line. It is visually apparent as the tilt of the line on a graph. The formula to calculate slope is based on how much \( y \) changes as \( x \) changes, given by: \[ m = \frac{\Delta y}{\Delta x} \] The "rise over run" approach helps visualize this:

• "Rise" refers to how much \( y \) increases or decreases.
• "Run" refers to the change in \( x \).

In this exercise, the slope \( m = 3 \) means that for every unit increase in \( x \), \( y \) increases by 3 units. Visualizing slope as a ratio between the change in \( y \) and \( x \), makes it easy to predict the behavior of the line across a graph. Understanding slope provides insight into the line’s direction:

• A positive slope indicates the line rises as it moves from left to right.
• A negative slope suggests the line falls as it moves from left to right.