Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the given information. (-1,3) and (4,-5)

Step-by-Step Solution

Answer

The line equation is \( y = \frac{-8}{5}x + \frac{7}{5} \).

Step 1: Understand Point-Slope Form

To find the equation of a line, we often use the point-slope form: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. We'll need the slope first, then we can write down the equation.

Step 2: Find the Slope

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points \((-1, 3)\) and \((4, -5)\), the slope is \( m = \frac{-5 - 3}{4 - (-1)} = \frac{-8}{5} \).

Step 3: Use Point-Slope Form

With the slope \(m = \frac{-8}{5}\) and using the point \((-1, 3)\), apply the point-slope form: \( y - 3 = \frac{-8}{5}(x - (-1)) \), which simplifies to \( y - 3 = \frac{-8}{5}(x + 1) \).

Step 4: Simplify to Slope-Intercept Form

Let's rearrange this equation to the slope-intercept form \(y = mx + b\): \( y - 3 = \frac{-8}{5}x - \frac{8}{5} \Rightarrow y = \frac{-8}{5}x - \frac{8}{5} + 3 \). Calculate further to get \( y = \frac{-8}{5}x + \frac{15}{5} - \frac{8}{5} \).

Step 5: Final Equation

Simplifying the right-hand side, we get \( y = \frac{-8}{5}x + \frac{7}{5} \). Therefore, the equation of the line in slope-intercept form is \( y = \frac{-8}{5}x + \frac{7}{5} \).

Key Concepts

Point-Slope Form

The point-slope form of a line is a convenient way to write the equation of a line when you have a point on the line and the slope. This form is expressed as:\[y - y_1 = m(x - x_1)\]Here, \((x_1, y_1)\) represents a known point on the line, and \(m\) is the slope of the line.
The point-slope form is particularly useful because it directly incorporates the slope and a point, making it straightforward to use when developing the equation of a line.
**Example** Let's consider the points (-1, 3) and (4, -5). Once you calculate the slope, you can insert one of these points into the point-slope form equation to find the line's equation. Using point (-1, 3), the point-slope equation becomes:\[y - 3 = \frac{-8}{5}(x - (-1))\]This clearly shows how the known point and the slope are seamlessly included in the formula.

Slope-Intercept Form

The slope-intercept form of a line is another popular way to express linear equations and is written as:\[y = mx + b\]In this equation, \(m\) is the slope and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
This form is widely used due to its simplicity and the ease with which you can interpret or graph a line directly from the equation.
After using the point-slope form, you can easily convert to the slope-intercept form. Continuing with the example of (-1, 3) and segment parallel to our calculated slope of \(\frac{-8}{5}\), we transition to the slope-intercept form as follows:\[y = \frac{-8}{5}x + \frac{7}{5}\]Here, the slope remains \(\frac{-8}{5}\) and \(\frac{7}{5}\) is the calculated y-intercept.

Slope Calculation

Calculating the slope of a line is a fundamental skill in algebra, crucial for forming the equation of a line. The slope measures how steep a line is and is calculated by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]**Steps to Calculate Slope**

• Identify two points on the line, say \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the y-coordinate of the first point from the y-coordinate of the second point: \((y_2 - y_1)\).
• Subtract the x-coordinate of the first point from the x-coordinate of the second point: \((x_2 - x_1)\).
• Divide the difference in the y-coordinates by the difference in the x-coordinates. This gives you the slope \(m\).

For example, with points (-1, 3) and (4, -5), the slope \(m\) is:\[m = \frac{-5 - 3}{4 - (-1)} = \frac{-8}{5}\]This negative slope indicates the line decreases as it moves from left to right, slanting downward.