Problem 22 - Elementary Algebra

Question

For Problems \(13-22\), find the equation of the line that contains the two given points. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. \((-2,0)\) and \((0,-9)\)

Step-by-Step Solution

Answer

The equation of the line is \(9x + 2y = -18\).

Step 1: Calculate the Slope

First, we need to find the slope of the line passing through the points \((-2, 0)\) and \((0, -9)\). The formula for the slope \(m\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plug in the values from the points: \(y_2 = -9\), \(y_1 = 0\), \(x_2 = 0\), and \(x_1 = -2\). Thus, we get: \[ m = \frac{-9 - 0}{0 - (-2)} = \frac{-9}{2} \].

Step 2: Use Point-Slope Form to Find the Line Equation

With the slope \(m = \frac{-9}{2}\), we can use the point-slope form of the line equation: \(y - y_1 = m(x - x_1)\). Use point \((-2, 0)\). Substitute \(m = \frac{-9}{2}\), \(x_1 = -2\), and \(y_1 = 0\) into the equation: \[ y - 0 = \frac{-9}{2}(x + 2) \], which simplifies to \( y = \frac{-9}{2}x - 9 \).

Step 3: Convert to Standard Form

We need the equation in the form \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers. Start from \(y = \frac{-9}{2}x - 9\). Multiply every term by 2 to eliminate the fraction: \[ 2y = -9x - 18 \]. Rearrange the terms to get the standard form: \[ 9x + 2y = -18 \].

Step 4: Verify Integer Coefficients and Adjust Signs

Check to ensure the coefficients \(A, B, C\) are integers and positive if possible (though not required more than one). Coefficients are \(9, 2, \) and \(-18\). These are integers and valid.

Key Concepts

Slope Calculation

The slope is a fundamental concept in geometry and algebra that shows the steepness or incline of a line. Calculating the slope between two points is the first step in determining the equation of a line.
To find the slope between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula calculates how much "rise", or vertical change, there is for a given "run", or horizontal change, between the points.

• In our example, the points \( (-2, 0) \) and \( (0, -9) \) are used to compute the slope.
• Plugging the coordinates into the formula: \( y_2 = -9, y_1 = 0, x_2 = 0, \) and \( x_1 = -2\), gives us \[m = \frac{-9 - 0}{0 - (-2)} = \frac{-9}{2}\].

This result, \( m = \frac{-9}{2} \), indicates the line descends since the slope is negative.
Now, understanding the slope helps visualize the trend of the line.

Point-Slope Form

Once the slope is known, point-slope form comes into play. This equation form conveniently combines the slope and a known point on the line. It's useful for expressing linear equations quickly.
The point-slope form is represented as:
\[y - y_1 = m(x - x_1)\]
This formula uses the slope \( m \) and any point on the line, \( (x_1, y_1) \), to find the equation of the line.

• For our previous data, with slope being \( \frac{-9}{2} \) and using the point \( (-2, 0)\):
• Substitute into the equation: \[ y - 0 = \frac{-9}{2}(x + 2) \]
• This simplication results in \[ y = \frac{-9}{2}x - 9 \]

Using the point-slope form is a straightforward method to establish the initial form of a line's equation. It readily allows for further transformation into other forms.

Standard Form

The standard form of a line's equation is widely used for its clean and integer-only appearance. It is typically structured as:
\[Ax + By = C\]
where \( A, B, \) and \( C \) are integers, and ideally, \( A \) should be non-negative.
To convert our point-slope equation to standard form, follow these steps:

• Start with the point-slope form result: \[ y = \frac{-9}{2}x - 9 \]
• Multiply every term by 2 to remove the fraction: \[ 2y = -9x - 18 \]
• Reorganize: \[ 9x + 2y = -18 \]

Now we have the standard form \( 9x + 2y = -18 \).
All coefficients are integers, satisfying the requirement. This finalized version provides clarity for equations of lines with integer values.