Question

Problems \(27-36\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find an equation for the line containing the points (-1,5) and (3,-3) . Write the equation using either the general form or the slope-intercept form, whichever you prefer.

Short answer

y = -2x + 3

Step-by-step solution

- Find the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \(x_2, y_2)\) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the points \((-1, 5)\) and \(3, -3)\), \[ m = \frac{-3 - 5}{3 - (-1)} = \frac{-3 - 5}{3 + 1} = \frac{-8}{4} = -2 \]

- Use Point-Slope Form

Using the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \]and using point \((-1, 5)\): \[ y - 5 = -2(x + 1) \]

- Simplify to Slope-Intercept Form

Distribute and simplify the equation to get it into the slope-intercept form \(y = mx + b)\): \[ y - 5 = -2x - 2 \]Adding 5 to both sides: \[ y = -2x + 3\]

Key concepts

Understanding the Slope

The slope of a line measures its steepness and direction. It is represented by the letter \(m\). The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the given points (-1, 5) and (3, -3), plug in the values into the formula:
\[ m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2 \].
Simple steps to calculate slope:

• Subtract the first y-value (\(y_1\)) from the second y-value (\(y_2\)).
• Subtract the first x-value (\(x_1\)) from the second x-value (\(x_2\)).
• Divide the result from step 1 by the result from step 2.

In this case, the slope is -2. This means the line falls 2 units vertically for every 1 unit it moves horizontally.

Using the Point-Slope Form

The point-slope form is a way to write the equation of a line when you know its slope and one of its points. The formula is:
\[ y - y_1 = m(x - x_1) \].
Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.
Given the point (-1, 5) and the slope -2, substitute these values into the formula:
\[ y - 5 = -2(x + 1) \].
The steps for using point-slope form:

• Identify a point (\(x_1, y_1\)) on the line and slope \(m\).
• Substitute the point's coordinates and the slope into the point-slope formula.

This results in the equation \(y - 5 = -2(x + 1)\).

Converting to Slope-Intercept Form

The slope-intercept form of the equation of a line is:
\[ y = mx + b \],
where \(m\) is the slope and \(b\) is the y-intercept. To convert from the point-slope to the slope-intercept form, follow these steps:

• Start with the point-slope equation: \(y - 5 = -2(x + 1)\).
• Distribute the slope -2: \(y - 5 = -2x - 2\).
• Add 5 to both sides to solve for \(y\): \( y = -2x + 3 \).

Now the equation is in slope-intercept form:
\( y = -2x + 3 \).
This means the line has a slope of -2 and crosses the y-axis at 3.

In summary:

• Point-slope form to slope-intercept form typically involves distributing and then isolating y.
• This form is very useful for quickly identifying the slope and y-intercept of a line.