Problem 24 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((-1,-2)\) and \((4,3)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = x - 1\).

Step 1: Identify Given Points

We are given two points through which the line passes: 1. First point: \((-1, -2)\)2. Second point: \((4, 3)\).

Step 2: Calculate the Slope (m) of the Line

The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points \[m = \frac{3 - (-2)}{4 - (-1)} = \frac{3 + 2}{4 + 1} = \frac{5}{5} = 1\]Therefore, the slope \(m\) of the line is 1.

Step 3: Use Point-Slope Form to Find Equation

The point-slope form of the equation of a line is \[y - y_1 = m(x - x_1)\]We can use either point \((-1, -2)\) or \((4, 3)\) along with the slope \(m = 1\).Using the first point \((-1, -2)\), the equation becomes:\[y + 2 = 1(x + 1)\]

Step 4: Simplify the Equation

Simplifying the equation from step 3:\[y + 2 = x + 1\]Reorganize the terms to convert it to slope-intercept form \(y = mx + b\): \[y = x + 1 - 2\]\[y = x - 1\]

Step 5: Verify with the Second Point

Ensure the equation \(y = x - 1\) holds for the second point \((4, 3)\): Substitute \(x = 4\) in the equation:\[y = 4 - 1 = 3\]The result \(3\) matches the given \(y\)-coordinate. Thus, the verified equation is correct.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness or incline. It helps us understand how much the line rises or falls as we move from one point to another. The formula to calculate the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

This formula computes the change in \( y \) (vertical change) divided by the change in \( x \) (horizontal change). Let's dive in through the example given:

• First point: \((-1, -2)\)
• Second point: \((4, 3)\)

Plugging these into the formula:

• \( m = \frac{3 - (-2)}{4 - (-1)} \)
• \( = \frac{3 + 2}{4 + 1} = \frac{5}{5} = 1\)

So, for our line, the slope \( m \) is \( 1 \). This means the line rises one unit up for every one unit it moves to the right.

Point-Slope Form

The point-slope form of a line's equation is one of the ways we can write the equation of a line when we know the slope of the line and one point on the line. The point-slope form is given by:

• \[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1)\) is a point on the line, and \( m \) is the slope. For our example:

• Slope \( m = 1 \)
• Choose the first point \((-1, -2)\)

Inserting these values gives:

• \( y + 2 = 1(x + 1) \)

This form is particularly useful because it directly shows us the slope, and using any point from the line lets us find an equation quickly. We can now convert this into other forms like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is a popular way of writing the equation of a line. It directly provides two key pieces of information: the line's slope and its intercept with the \( y \)-axis. The formula is expressed as:

• \[ y = mx + b \]

Here, \( m \) represents the slope, and \( b \) is the \( y \)-intercept where the line crosses the \( y \)-axis. To convert our point-slope equation to slope-intercept form:

• Start from: \( y + 2 = x + 1 \)
• Solve for \( y \): \( y = x + 1 - 2 \)
• Result: \( y = x - 1 \)

In this case, the slope \( m \) is \( 1 \) and the intercept \( b \) is \( -1 \). This format makes it easy to graph the line, since you can start at the \( y \)-intercept and use the slope to find other points on the line.

Verifying Solutions

Verifying a solution ensures that our derived equation satisfies all given conditions or points. Check your equation against alternate points not used in forming the equation. For our example, we verify the line's equation \( y = x - 1 \) using the second point \((4, 3)\).

• Substitute \( x = 4 \) into the equation: \( y = 4 - 1 \)
• Calculate: \( y = 3 \)

This matches the \( y \)-coordinate of our second point. Thus, the equation is indeed correct and valid. Verifying is an essential step as it confirms the line accurately represents the points given, avoiding potential errors in calculations.