Question

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 of the line that contains the points (0,1) and \((8,-4) .\) Write the equation in general form.

Short answer

The equation of the line is \( 5x + 8y - 8 = 0 \).

Step-by-step solution

- Determine the Slope

The slope (m) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, we have points \( (0,1) \) and \( (8,-4) \). So, \[ m = \frac{-4 - 1}{8 - 0} = \frac{-5}{8} \]

- Use Point-Slope Form

The point-slope form of a line's equation is given by: \[ y - y_1 = m(x - x_1) \] Using the point \( (0, 1) \) and slope \( m = -\frac{5}{8} \), the equation becomes: \[ y - 1 = -\frac{5}{8}(x - 0) \] Simplify this to: \[ y - 1 = -\frac{5}{8}x \]

- Convert to General Form

The general form of a linear equation is \( Ax + By + C = 0 \). From the simplified equation, \[ y - 1 = -\frac{5}{8}x \] Rearrange to get rid of the fraction: \[ 8(y - 1) = -5x \] \[ 8y - 8 = -5x \] Add \( 5x \) to both sides to get: \[ 5x + 8y - 8 = 0 \]

Key concepts

Slope

Understanding the slope of a line is crucial as it tells us how steep the line is and in which direction it slants. The slope is usually represented as 'm' and can be calculated when you have two points. For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope m is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• If the value of m is positive, the line rises from left to right.
• If it is negative, the line falls from left to right.
• If the value is zero, the line is horizontal.
• If the slope is undefined (division by zero), the line is vertical.

In our example, the points given are \((0, 1)\) and \((8, -4)\). Plugging these into the formula, we get:
\[ m = \frac{-4 - 1}{8 - 0} = \frac{-5}{8} \]
So, our slope, m, is \(-\frac{5}{8} \). This means the line falls as we move from left to right.

Point-Slope Form

The point-slope form is particularly useful when you know a point on the line and the slope. The formula for the point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1)\) is a point on the line, and m is the slope. In our example, we use the point \((0, 1)\) and the slope \(-\frac{5}{8}\) that we calculated earlier. Plugging these values into the formula, we get:
\[ y - 1 = -\frac{5}{8}(x - 0) \]
Which simplifies to:
\[ y - 1 = -\frac{5}{8}x \]
This is our equation in point-slope form. It's handy because it's straightforward and lets you easily see the slope of the line.

General Form

The general form of a line's equation is written as:
\ Ax + By + C = 0 \
To convert our equation from the point-slope form to the general form, we start with:
\[ y - 1 = -\frac{5}{8}x \]
First, eliminate the fraction by multiplying through by 8:
\[ 8(y - 1) = -5x \]
Which simplifies to:
\[ 8y - 8 = -5x \]
Rearrange all terms to one side to fit the general form template:
\[ 5x + 8y - 8 = 0 \]
This is our line's equation in general form. The coefficients \(A, B, \text{and} C\) now represent our equation concisely:

• A = 5
• B = 8
• C = -8

The general form makes it easy to see how the line compares to other lines and accommodates different algebraic operations effectively.