Problem 71 - Calculus Volume 1

Question

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. Passing through \((2,1)\) and \((-2,-1)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{2}x \).

Step 1: Identify the Slope Formula

The slope-intercept form of a line’s equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. First, we need to find the slope (\( m \)). The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Substitute Values into the Slope Formula

Plug the given points \((2,1)\) and \((-2,-1)\) into the slope formula: \( m = \frac{-1 - 1}{-2 - 2} \). Simplify the expression to find the slope.

Step 3: Calculate the Slope

Perform the calculations: \( m = \frac{-2}{-4} = \frac{1}{2} \). The slope of the line is \( \frac{1}{2} \).

Step 4: Use the Point-Slope Form to Find the Y-Intercept

The point-slope form of a line’s equation is \( y - y_1 = m(x - x_1) \). Use point \((2,1)\) and slope \( \frac{1}{2} \): \( y - 1 = \frac{1}{2}(x - 2) \).

Step 5: Convert to Slope-Intercept Form

Simplify the equation \( y - 1 = \frac{1}{2}(x - 2) \) to get \( y = \frac{1}{2}x - 1 + 1 \). This simplifies further to \( y = \frac{1}{2}x \).

Step 6: Conclusion

The slope-intercept form of the line passing through the points \((2,1)\) and \((-2,-1)\) is \( y = \frac{1}{2}x \).

Key Concepts

Slope-Intercept Form

One of the most common ways to express the equation of a line is through the slope-intercept form. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis. This form is especially helpful because it provides direct information about the line's direction and y-intercept.

• **Slope**: The value of \( m \) tells us how steep the line is, as well as its direction. A positive slope means the line is increasing, while a negative slope indicates a decreasing line.
• **Y-Intercept**: The coefficient \( b \) gives the y-coordinate at which the line crosses the y-axis. This means if \( x = 0 \), then \( y = b \).

The slope-intercept form is very flexible and can be easily converted to or from other forms. Once you calculate the slope using the slope formula and know a point on the line, you can rearrange the equation to find \( b \) and complete the equation.

Point-Slope Form

The point-slope form is particularly useful for quickly writing equations of lines when you know one point on the line and its slope. This form looks like \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) are the coordinates of the point through which the line passes.

• **Derived information**: Unlike slope-intercept form, we begin here with the slope \( m \) and a specific point to find the equation of the line.
• **Flexibility**: It's great for developing an equation quickly, especially when only a point and a slope are given.

By using the point-slope form, you can simply substitute the given values, manipulate the equation algebraically, and convert it into the more universally recognized slope-intercept form to find further insights, such as the y-intercept or to graph the line directly.

Slope Formula

The slope formula defines the steepness and the direction of a line between two points. Given two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), the formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• **Difference in y-coordinates**: The numerator \( y_2 - y_1 \) determines the change in the vertical direction.
• **Difference in x-coordinates**: The denominator \( x_2 - x_1 \) signifies the change in the horizontal direction.

The slope formula is pivotal in finding the exact slope \( m \) of the line, which gives critical information such as whether the line rises or falls as it moves from left to right. Once you calculate this slope, you can plug it into other forms like the point-slope or slope-intercept form to obtain a comprehensive understanding of the line's equation.