Problem 17 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (6,5) and (7,1)

Step-by-Step Solution

Answer

Answer: The equation of the line in point-slope form is \(y-5 = -4(x-6)\).

Step 1: Find the slope of the line.

To find the slope of the line passing through two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\), we use the slope formula: \(m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). In this case, we are given the points (6,5) and (7,1). Plugging in the points, we get the slope, \(m\): $$ m = \frac{1-5}{7-6} = \frac{-4}{1} = -4 $$

Step 2: Write the point-slope form equation.

Now that we have the slope \(m=-4\), we can choose either of the given points and plug it into the point-slope form equation. Let's choose point (6,5). Using the point-slope formula \(y - y_{1} = m(x- x_{1})\), we substitute the values of \(m, x_{1}\), and \(y_{1}\): $$ y-5 = -4(x-6) $$ This is the equation of the line in point-slope form passing through the points (6,5) and (7,1).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells us how much y changes for a change in x. When you have two points on a line, such as

• (6, 5)
• (7, 1)

you can find the slope using the formula: \\( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For this pair of points, the slope \( m \) is calculated as follows:

• Subtract the y-coordinates: \(1 - 5 = -4\)
• Subtract the x-coordinates: \(7 - 6 = 1\)

So, \\( m = \frac{-4}{1} = -4 \).

A negative slope indicates that the line is going downwards as it moves from left to right. In our case, the slope \( -4 \) tells us that for each unit increase in x, y decreases by 4 units. Understanding slope is crucial as it lays the foundation for writing equations of lines in various forms.

Equation of a Line

Once we have the slope of a line, the next step is to write the equation of the line. There are different forms for the equation of a line, but one of the most useful is the **point-slope form**. The formula is:\( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope, and \((x_1, y_1)\) is a point on the line. For our exercise, \( m = -4 \) and we can use point (6,5). Placing these into the formula gives us:

• \( y - 5 = -4(x - 6) \)

This is the point-slope form equation of the line passing through the points (6,5) and (7,1).

This equation form is very powerful because it immediately gives you the slope and a point on the line, making it easier to graph or further manipulate into other forms like slope-intercept form.

Mathematical Problem Solving

Solving mathematical problems involves a step-by-step process where each step builds upon the previous one. In the context of writing equations of lines, consider the sequence:

• Identify given data
• Use formulas to calculate necessary values
• Substitute these values into relevant equations

For instance, when asked to find a line's equation through two points, you might first find the slope. Then, choose a formula, like point-slope, to write the equation.

This process helps in organizing your calculations and ensuring accuracy. It's crucial to understand each step fully, as mastering these fundamentals will enhance your problem-solving skills not just in geometry, but in all areas of mathematics. Emphasizing clarity and precision in each step aids in avoiding errors and understanding the concepts in depth.