Problem 5 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { The slope is } 4 \text { and through the point }(2,-3) \text {. } $$

Step-by-Step Solution

Answer

\( y = 4x - 11 \)

Step 1: Identify the Point-Slope Form

The point-slope form of the equation of a line is: \( y - y_1 = m(x - x_1) \) where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Step 2: Substitute Given Values

Substitute the given slope \( m = 4 \) and the point \( (x_1, y_1) = (2, -3) \) into the point-slope formula: \( y - (-3) = 4(x - 2) \)

Step 3: Simplify the Equation

Simplify the equation by distributing and then solving for \( y \): \( y + 3 = 4(x - 2) \)\( y + 3 = 4x - 8 \)Subtract 3 from both sides: \( y = 4x - 8 - 3 \)\( y = 4x - 11 \)

Key Concepts

Point-Slope Form

The point-slope form is a special way to write the equation of a line. It is particularly useful when you know one point on the line and the slope. The point-slope form is written as:

• y - y_1 = m(x - x_1)

In this formula, \(y_1\) and \(x_1\) are the coordinates of the point on the line. The letter \(m\) represents the slope of the line. When you insert these values into the formula, you can describe the equation of the line. This form is helpful because it directly incorporates the slope and a single point, making it straightforward to plug in values and find the equation of your line.

Slope

The slope of a line is a measure of how steep the line is. Slope is often represented by the letter \(m\). The slope tells you how much the y-coordinate of a point on the line changes for a given change in the x-coordinate. The formula for finding slope when you have two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

In simpler terms, it is 'rise over run': how far you go up or down for each step you take to the right. A positive slope means the line is increasing, a negative slope means the line is decreasing, a zero slope indicates a horizontal line, and an undefined (or infinite) slope describes a vertical line.

Linear Equations

Linear equations are equations of the first degree. This means the variables are only to the power of one. The general form of a linear equation in the coordinate plane is:

• Ax + By = C

where A, B, and C are constants. The slope-intercept form, which is another way to write linear equations, is:

• y = mx + b

Here, \(m\) is the slope and \(b\) is the y-intercept, which is where the line crosses the y-axis. Linear equations graph as straight lines. They are frequently used in algebra and everyday problem-solving because they describe constant rates of change.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This modern method allows us to describe geometric shapes algebraically and solve geometric problems using equations. The coordinate system consists of two axes: the x-axis (horizontal) and y-axis (vertical). Every point in the plane is described by an ordered pair \((x, y)\), which defines its position relative to the origin (0,0). Using coordinate geometry, you can:

• Find the distance between two points.
• Determine the midpoint of a segment.
• Write the equations of lines, circles, and other shapes.
• Analyze the slopes and intercepts of lines.

This makes it a powerful tool for linking algebraic equations with geometric visualization.