Problem 6 - Applied Calculus

Question

Find an equation for the line that passes through the given points. $$ (0,0) \text { and }(1,1) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Points

The given points are \((0, 0)\) and \((1, 1)\). These will be used to find the slope of the line and to write the equation of the line.

Step 2: Find the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 1\), \(y_2 = 1\) to find the slope: \( m = \frac{1 - 0}{1 - 0} = 1 \).

Step 3: Use the Point-Slope Form

The point-slope form of a line equation is \( y - y_1 = m(x - x_1) \). Using point \((0, 0)\) and the slope \(m = 1\), the equation becomes: \( y - 0 = 1(x - 0) \) or simply \( y = x \).

Step 4: Write the Final Equation

After simplifying, the equation of the line that passes through the given points is \( y = x \). This is the final equation of the line.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. Imagine a hill or a slide; the slope tells you how steep it is. In mathematics, the slope is represented by the letter "m" and is calculated using two points that a line passes through. The formula to determine the slope when you have two points, \[ (x_1, y_1) \text{ and } (x_2, y_2), \]is:\[ m = \frac{y_2 - y_1}{x_2 - x_1}. \]Let’s break this down:

• \(y_2 - y_1\) represents the change in the vertical direction.
• \(x_2 - x_1\) signifies the change in the horizontal direction.

This ratio tells you how much you move up (or down) for every unit you move to the right. If the result is positive, the line slants upwards; if it's negative, the line slants downwards. An example is the line passing through the points (0,0) and (1,1), which results in a slope of 1, meaning it rises 1 unit up for every unit it runs to the right.

Point-Slope Form

The point-slope form helps in creating the equation of a line when you know the slope and at least one point on the line. The point-slope formula is represented as:\[ y - y_1 = m(x - x_1), \]where

• \(m\) is the slope,
• \((x_1, y_1)\) is the given point on the line.

To use this formula effectively, let's take a look at an example: Consider you have a slope \(m = 1\) and a point \((0, 0)\) on the line. By plugging in these values into the point-slope formula, you get:\[ y - 0 = 1(x - 0). \]This simplifies to:\[ y = x, \]which is the equation of the line. The point-slope form is especially useful because it directly incorporates a point on the line and the slope, making it a powerful tool for deriving the equation quickly.

Equation of a Line

The equation of a line represents all the points that lie on that line in a coordinate plane. It tells you exactly how the line behaves across the grid.There are various forms to write this equation, each suitable for different situations. One of the most straightforward forms is the slope-intercept form, which is:\[ y = mx + b, \]where:

• \(m\) is the slope of the line,
• \(b\) is the y-intercept, or where the line crosses the y-axis.

For the exercise with points (0,0) and (1,1), the equation turns out to be \(y = x\). This simple equation means that for each value of \(x\), \(y\) is equal—this reflects a perfect balance, moving diagonally across the graph.Understanding equations of lines is crucial for identifying and interpreting various lines in geometry and algebra, providing a foundational skill for more complex math topics.