Problem 35 - College Algebra

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

Step 1: Calculate the Slope

To find the equation of the line, we first need the slope of the line passing through the points \((-1, 3)\) and \((4, -5)\). The formula for the slope \(m\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substituting the given points, we have \(y_2 = -5\), \(y_1 = 3\), \(x_2 = 4\), and \(x_1 = -1\). So, \[ m = \frac{-5 - 3}{4 - (-1)} = \frac{-8}{5} = -\frac{8}{5} \].

Step 2: Use Point-Slope Form

With the slope found, we can use the point-slope form of the equation of the line, which is \( y - y_1 = m(x - x_1) \). We can choose one of the given points, say \((-1, 3)\), to use in the formula. Substituting \(m = -\frac{8}{5}\), \(x_1= -1\), and \(y_1 = 3\), the equation becomes: \[ y - 3 = -\frac{8}{5}(x + 1) \].

Step 3: Simplify the Equation

Now, we simplify the equation: \( y - 3 = -\frac{8}{5}(x + 1) \). First, distribute the slope: \[ y - 3 = -\frac{8}{5}x - \frac{8}{5} \]. Then, add 3 to both sides to solve for \(y\): \[ y = -\frac{8}{5}x - \frac{8}{5} + 3 \]. Convert 3 to a fraction with a denominator of 5: \(3 = \frac{15}{5}\). Therefore, \[ y = -\frac{8}{5}x - \frac{8}{5} + \frac{15}{5} \]. Combine like terms: \[ y = -\frac{8}{5}x + \frac{7}{5} \].

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness, and it's an essential part of finding a linear equation. To calculate the slope, you need two points through which the line passes. Let's consider the points

• \((-1, 3)\)
• \((4, -5)\)

To find the slope \((m)\), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates of your points into this formula. Here,

• \(y_2 = -5\), \(y_1 = 3\)
• \(x_2 = 4\), \(x_1 = -1\)

Plug them in, and you'll get:\[ m = \frac{-5 - 3}{4 - (-1)} = \frac{-8}{5} \]So, the slope of this line is \(-\frac{8}{5}\). This negative value indicates that the line descends from left to right.

Point-Slope Form

Point-slope form is a quick way of writing the equation of a line. It helps us understand how the line behaves and passes through a specific point with a particular slope.The point-slope form of a linear equation is expressed as:\[ y - y_1 = m(x - x_1) \]Here's what each symbol means:

• \(y_1\) and \(x_1\): The coordinates of a point on the line
• \(m\): The slope of the line

For our example, using the point \((-1, 3)\) and the slope \(-\frac{8}{5}\), the form becomes:\[ y - 3 = -\frac{8}{5}(x + 1) \]This form is handy because it's derived directly from the relationship between the change in \(y\) and the change in \(x\) across two points on the line.

Linear Equations

Linear equations describe straight lines on a graph and are fundamental in algebra. They are generally expressed in the form:\[ y = mx + c \]where \(m\) is the slope, and \(c\) is the y-intercept—the point where the line crosses the y-axis.In the step-by-step solution, after setting up the point-slope form, we simplified it to get:\[ y = -\frac{8}{5}x + \frac{7}{5} \]To achieve this:

• Distribute the slope: \( y - 3 = -\frac{8}{5}x - \frac{8}{5} \)
• Add 3 to both sides to solve for \(y\): \( y = -\frac{8}{5}x - \frac{8}{5} + 3 \)
• Convert 3 to a fraction: \(3 = \frac{15}{5}\)
• Combine like terms: \( y = -\frac{8}{5}x + \frac{7}{5} \)

Now, the equation \(y = -\frac{8}{5}x + \frac{7}{5}\) is in slope-intercept form. It tells you everything about the line: how steep it is and where on the y-axis it begins.