Problem 23 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((2,1)\) and \((1,6)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -5x + 11\).

Step 1: Find the Slope

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the given points \((2,1)\) and \((1,6)\), we have:\[ m = \frac{6 - 1}{1 - 2} = \frac{5}{-1} = -5 \]So, the slope of the line is \(-5\).

Step 2: Use Point-Slope Form

The point-slope form of the equation of a line is given by:\[ y - y_1 = m(x - x_1) \]We can use either of the given points; let's use \((2,1)\).Using the slope from Step 1 (\(m = -5\)) and the point \((2,1)\), the equation becomes:\[ y - 1 = -5(x - 2) \]

Step 3: Simplify to Slope-Intercept Form

Now, solve the equation from Step 2 to put it in slope-intercept form \(y = mx + b\).Starting with:\[ y - 1 = -5(x - 2) \]Distribute the \(-5\):\[ y - 1 = -5x + 10 \]Add 1 to both sides to solve for \(y\):\[ y = -5x + 11 \]This is the equation of the line.

Key Concepts

Slope of a Line

The slope of a line is a number that describes how steep the line is. It tells us how much the y-value changes for a change in the x-value. Imagine a hill: the steeper it is, the greater the slope. To find the slope when you have two points, we use the formula:

• Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

This formula calculates the change in y (vertical change) divided by the change in x (horizontal change). For the points \((2, 1)\) and \((1, 6)\), the slope calculation is:

• \( m = \frac{6 - 1}{1 - 2} = \frac{5}{-1} = -5 \)

This means that for each unit increase in x, y decreases by 5 units, indicating a downward sloping line.

Point-Slope Form

The point-slope form is a great way to write the equation of a line when you know the slope and any point on the line. The formula is:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \)

This form emphasizes the slope \(m\), and \((x_1, y_1)\) which is a specific point on the line. To use the point-slope formula:

• Select one point on the line, for example \((2, 1)\).
• Substitute \(m = -5\) and \((x_1, y_1) = (2, 1)\) into the formula: \( y - 1 = -5(x - 2) \).

This results in an equation that represents the line passing through the given point with the given slope.

Slope-Intercept Form

The slope-intercept form is probably the most popular way to write an equation of a line because it clearly shows both the slope \(m\) and the y-intercept \(b\). This form is written as:

• Slope-Intercept Form: \( y = mx + b \)

When you want to rewrite a point-slope equation into the slope-intercept form, you focus on solving for \(y\). Let's work from the equation \( y - 1 = -5(x - 2) \):

• Start by distributing \(-5\): \( y - 1 = -5x + 10 \).
• Add 1 to both sides to isolate \(y\): \( y = -5x + 11 \).

Now, the equation \( y = -5x + 11 \) shows that the line has a slope of \(-5\) and crosses the y-axis at \(11\). This clear form helps us quickly graph the line or understand its direction and position.