Problem 42 - Brief Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Passing through the points \((3,-1)\) and \((6,0)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{3}x - 2\).

Step 1: Identify the Slope Formula

The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Substitute the Given Points into the Slope Formula

Substitute the points \((3, -1)\) and \((6, 0)\) into the formula: \(m = \frac{0 - (-1)}{6 - 3} = \frac{1}{3}\).

Step 3: Use the Point-Slope Form of the Equation

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). We'll use the point \((3, -1)\) and the slope we just calculated \(m = \frac{1}{3}\).

Step 4: Write the Equation in Point-Slope Form

Substitute the slope and the point into the form: \(y + 1 = \frac{1}{3}(x - 3)\).

Step 5: Convert to Slope-Intercept Form

Solve the equation in point-slope form to find \(y\) in terms of \(x\): \(y + 1 = \frac{1}{3}x - 1\). Then, simplify to \(y = \frac{1}{3}x - 2\) by subtracting 1 from each side.

Key Concepts

Slope Formula

The slope formula is an essential tool in geometry, especially when it comes to lines and linear equations. The slope, denoted as \( m \), defines how steep or flat a line is on a graph. It's calculated as the change in the y-values (vertical change) divided by the change in x-values (horizontal change) between two points on the line. This formula is expressed as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here’s a handy way to remember: rise over run. "Rise" is how much you move up or down, and "run" is how much you move left or right. For example, using the points \((3, -1)\) and \((6, 0)\), you plug in these coordinates to find the slope:

• Rise: \(0 - (-1) = 1\)
• Run: \(6 - 3 = 3\)

Thus, the slope \( m \) is \( \frac{1}{3} \). This means for every three units you go to the right (run), you go one unit up (rise). Understanding this concept is crucial as it lays the foundation for how we determine the direction and steepness of a line on a coordinate plane.

Point-Slope Form

Once you've got the slope, point-slope form is a very helpful way to write the equation of a line, especially when you know a point on the line and the slope. It might look complicated, but it’s pretty straightforward:\[y - y_1 = m(x - x_1)\]In this equation, \((x_1, y_1)\) is any point on the line (you can pick your favorite!), and \( m \) is the slope we’ve already calculated. For our specific exercise, we used the point \((3, -1)\) and the slope \( \frac{1}{3} \). Here’s how it works:

• Insert the slope: \( \frac{1}{3} \)
• Insert the point: \((3, -1)\)

The line equation becomes:\[y + 1 = \frac{1}{3}(x - 3)\]This form is very useful, as it easily adapts to different situations just by changing the point \((x_1, y_1)\). Once you understand how to place a point and a slope into this formula, finding any line's equation becomes much more approachable.

Slope-Intercept Form

The slope-intercept form is probably the most recognized way to write the equation of a line, especially because it includes the line's slope and its y-intercept (the point where the line crosses the y-axis). The general formula is:\[y = mx + b\]Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. Transitioning from point-slope form to this form helps visualize the line's behavior on a graph quickly. For the line passing through the points \((3, -1)\) and \((6, 0)\), we convert from point-slope form:

• Start with \( y + 1 = \frac{1}{3}(x - 3) \)
• Distribute the slope: \( y + 1 = \frac{1}{3}x - 1 \)
• Solve for \( y \): subtract 1 from both sides to get \( y = \frac{1}{3}x - 2 \)

This final equation, \( y = \frac{1}{3}x - 2 \), is in slope-intercept form. It shows that the line crosses the y-axis at \(-2\) and rises one unit for every three units it goes to the right. It’s a very common and practical way to express equations in algebra and can help easily plot the line on a graph.