Problem 12 - Calculus Single Variable

Question

Write the point-slope equation of the line determined by the two given points. (1,-5),(-5,1)

Step-by-Step Solution

Answer

y + 5 = -1(x - 1)

Step 1: Determine the Slope

First, we need to find the slope of the line that passes through the two given points \((1, -5)\) and \((-5, 1)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting in our values, we get:\[m = \frac{1 - (-5)}{-5 - 1} = \frac{1 + 5}{-6} = \frac{6}{-6} = -1\]The slope \(m\) of the line is \(-1\).

Step 2: Use the Point-Slope Form

The point-slope form of a line's equation is given by:\[y - y_1 = m(x - x_1)\]where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. We can use either of the given points to write the equation. Let's use the point \((1, -5)\).Substituting the slope \(m = -1\) and the point \((x_1, y_1) = (1, -5)\) into the formula gives:\[y + 5 = -1(x - 1)\]This is the point-slope form of the equation for the line.

Step 3: Simplify the Equation (Optional)

For clarity, we can simplify the point-slope equation to the slope-intercept form \(y = mx + b\) if needed.Starting from:\[y + 5 = -1(x - 1)\]Distribute \(-1\) on the right side:\[y + 5 = -x + 1\]Isolate \(y\) by subtracting 5 from both sides:\[y = -x + 1 - 5\]\[y = -x - 4\]This is the slope-intercept form of the line's equation. However, the exercise specifically asks for the point-slope form, so our answer is:\[y + 5 = -1(x - 1)\]

Key Concepts

Slope Formula

When we want to find the equation of a line using two given points, the first thing we need is the slope of the line. We calculate the slope using the Slope Formula. This formula helps us determine how steep the line is, essentially measuring the "rise over run". For any two points

• Given as
  • \((x_1, y_1)\) and
  • \((x_2, y_2)\)
• The Slope Formula is
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
  
  
This formula tells us how much the line goes up or down (the change in \(y\)) for every step it moves left or right (the change in \(x\)). It is crucial for establishing the direction and position of the line on the graph. In our example, applying this formula gives us a slope \(m = -1\), which indicates a downward slope or a decrease by 1 unit in \(y\) for every unit increase in \(x\). Understanding and calculating the slope correctly lays the groundwork for constructing the line's equation.

Equation of a Line

The Equation of a Line represents all of the points that lie on the line in a mathematical form. Once we have the slope using the slope formula, we can construct the line's equation in various forms. The first step is often determining which form of the equation is required.

• There are several forms to express the equation of a line:
• We can use Point-Slope Form when a point and the slope are known.
• The Slope-Intercept Form is great when you have or want to find the y-intercept easily.

Each form has its advantages, depending on what information is available and what you wish to find out. Our example used the Point-Slope Form because we were given points through which the line passes, and we calculated the slope. Understanding which form to use is a vital skill for effectively solving problems involving lines.

Slope-Intercept Form

The Slope-Intercept Form is one of the most common ways to present the Equation of a Line, thanks to its simplicity and ease of interpretation. It's written as:

• \(y = mx + b\)
  • Where "\(m\)" is the slope of the line
  • "\(b\)" is the y-intercept, which is where the line crosses the y-axis

This form is particularly useful because it gives immediate insight into the steepness (slope) and the starting point (y-intercept) of the line, making it easier to graph. Although our task required us to find the point-slope form, converting to slope-intercept form can provide a clearer picture of the line's behavior. In the example solution, simplifying to this form resulted in \(y = -x - 4\). It represents the same line, but in a way that highlights its crossing point on the y-axis, which is \((-4)\). Understanding and using the slope-intercept form is a valuable skill in many mathematical and real-world applications.