Problem 22 - Elementary Algebra

Question

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. (Objective \(1 \mathrm{~b}\) ) \((-2,0)\) and \((0,-9)\)

Step-by-Step Solution

Answer

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

Step 1: Calculate the Slope

First, we need to calculate the slope \( m \) of the line that passes through the two given points \((-2,0)\) and \((0,-9)\). The formula for the slope \( m \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the given points: \[ m = \frac{-9 - 0}{0 - (-2)} = \frac{-9}{2} \]

Step 2: Write the Point-Slope Equation

Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), we substitute one of the points, for example \((-2,0)\), and the slope \( m = \frac{-9}{2} \):\[ y - 0 = \frac{-9}{2}(x + 2) \]Simplifying gives:\[ y = \frac{-9}{2}x - 9 \]

Step 3: Convert to Standard Form

We need to convert the slope-intercept form \( y = \frac{-9}{2}x - 9 \) into the standard form \( Ax + By = C \). To eliminate the fraction, multiply every term by 2:\[ 2y = -9x - 18 \]Rearrange to get the standard form:\[ 9x + 2y = -18 \]

Key Concepts

Slope Calculation

When finding the equation of a line, the slope is a key component. The slope, often denoted as \( m \), indicates the steepness of the line, or how much the line inclines or declines. It is calculated using two points on the line, typically represented as \((x_1, y_1)\) and \((x_2, y_2)\). The formula for calculating slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the points \((-2, 0)\) and \((0, -9)\), the calculation becomes:\[ m = \frac{-9 - 0}{0 - (-2)} = \frac{-9}{2} \]This result tells us that for each step of 2 units to the right on the x-axis, the y-value decreases by 9 units. Understanding the slope is critical for writing the equation of the line in various forms.

Point-Slope Form

The point-slope form of an equation is one of the simplest ways to write the equation of a line when given a point and the slope. The formula for the point-slope form is:\[ y - y_1 = m(x - x_1) \]This form uses a single point \((x_1, y_1)\) on the line and the line's slope \( m \). Using the point \((-2, 0)\) and the slope \( m = \frac{-9}{2} \), the equation is:\[ y - 0 = \frac{-9}{2}(x + 2) \]Simplifying this equation gives us:\[ y = \frac{-9}{2}x - 9 \]Point-slope form is particularly useful for writing the equation quickly and for converting it into other forms, like the slope-intercept or standard form.

Standard Form of a Linear Equation

The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A, B, \) and \( C \) are integers, and \( A \) should be non-negative. Often, converting from a slope-intercept or point-slope form to standard form involves manipulating the equation to eliminate fractions and decimals.From the slope-intercept form \( y = \frac{-9}{2}x - 9 \), multiplying every term by 2 clears the fraction, yielding:\[ 2y = -9x - 18 \]Rearranging gives us:\[ 9x + 2y = -18 \]This is the desired standard form, ensuring all coefficients are integers and providing a neat, uniform way to represent linear equations. It is widely used because of its clear specification of the relationship between \( x \) and \( y \). Understanding each form helps in analyzing and graphing linear equations easily.