Problem 68 - Calculus Volume 1

Question

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. $$ =3, \text { passes through }(-3,2) $$

Step-by-Step Solution

Answer

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

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. Our task is to find the equation of the line in this format.

Step 2: Identify Given Information

From the problem, we know the slope \( m = 3 \) and that the line passes through the point \((-3, 2)\). These pieces of information will help us determine the y-intercept \( b \) to complete the equation.

Step 3: Use Point-Slope Formula

The point-slope formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line. Substituting the point \((-3, 2)\) and slope \( m = 3 \) into this formula, we get: \[ y - 2 = 3(x + 3) \]

Step 4: Simplify the Equation

Expand and simplify the equation from Step 3:1. Distribute the 3: \( y - 2 = 3x + 9 \).2. Add 2 to both sides to isolate \( y \): \( y = 3x + 11 \).

Step 5: Verify the Solution

To verify, plug the point \((-3, 2)\) back into the equation \( y = 3x + 11 \):- When \( x = -3 \), \( y = 3(-3) + 11 = -9 + 11 = 2 \).The calculation is correct, confirming that the equation is accurate.

Key Concepts

Equation of a Line

Understanding the equation of a line is crucial in algebra, especially when you're dealing with linear equations. The most common way to express a line is in **slope-intercept form**, given by the formula: \( y = mx + b \)
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept—the point where the line crosses the y-axis. This format is incredibly useful because it allows you to quickly identify both the slope and the point at which the line intersects the y-axis.

To find the equation of a line, you generally need two pieces of information:

• The slope \( m \)
• A point the line passes through, often provided as coordinates \( (x_1, y_1) \)

These elements can help you construct the line's equation in slope-intercept form.

Point-Slope Formula

The point-slope formula is a handy tool for finding the equation of a line when you know the slope and one point on the line. It is expressed as:

\( y - y_1 = m(x - x_1) \)

This formula uses:

• \( m \) - the slope of the line
• \( (x_1, y_1) \) - a known point on the line

To apply the point-slope formula, take these steps:

• Substitute the slope \( m \) into the formula
• Use the provided point \( (x_1, y_1) \) and substitute these values in as well
• Solve for \( y \) to express the equation in slope-intercept form

This conversion involves simplifying the expression to get \( y = mx + b \). Once you have this form, you've identified both the slope and y-intercept.

Y-Intercept

The **y-intercept** is a fundamental part of a linear equation, represented by \( b \) in the slope-intercept form \( y = mx + b \). It denotes the exact point where the line crosses the y-axis, expressed as the coordinate \( (0, b) \). Understanding the y-intercept is essential because it helps locate the starting point of the line on the y-axis.

To find the y-intercept when given a slope and a point that the line passes through, follow these steps:

• Start with the point-slope formula: \( y - y_1 = m(x - x_1) \)
• Substitute the slope \( m \) and the given point \( (x_1, y_1) \)
• Solve for \( y \) to put it in slope-intercept form and identify \( b \)

This process allows you to transition from knowing just a point and a slope to having the full equation, complete with its y-intercept, making it easier to graph or understand the line's behavior.