Problem 18 - 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. \((-8,-7)\) and \((-3,-1)\)

Step-by-Step Solution

Answer

The line equation is \( 6x - 5y = -13 \).

Step 1: Calculate the Slope

To find the slope (m) of the line that connects the points (-8,-7) and (-3,-1), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the values from the points:\[ m = \frac{-1 + 7}{-3 + 8} = \frac{6}{5} \] So the slope is \( \frac{6}{5} \).

Step 2: Use Point-Slope Form

Next, use the point-slope formula, which is given as:\[ y - y_1 = m(x - x_1) \]Substitute one of the points, say (-8,-7), and the slope \( \frac{6}{5} \):\[ y + 7 = \frac{6}{5}(x + 8) \].

Step 3: Convert to Standard Form

Distribute the slope on the right side:\[ y + 7 = \frac{6}{5}x + \frac{48}{5} \]Multiply the entire equation by 5 to clear the fraction:\[ 5y + 35 = 6x + 48 \]Rearrange to get the equation in standard form \( Ax + By = C \):\[ 6x - 5y = -13 \] (by subtracting \(5y + 35\) from both sides and simplifying).

Step 4: Verify Integers in the Equation

Ensure the equation is \( 6x - 5y = -13 \), and all coefficients \( A, B, C \) are integers.

Key Concepts

Slope Calculation

Finding the slope of a line is a fundamental step when working with line equations. The slope tells us how steep the line is and the direction it tilts. We often label the slope as 'm'. It is calculated using two points on the line, denoted \(x_1, y_1\) and \(x_2, y_2\). The formula to find the slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the 'rise' over the 'run' between the two points: how much it goes up as it goes sideways.For example, using the points \(x_1, y_1\) = (-8,-7) and \(x_2, y_2\) = (-3,-1), you plug these values into the formula:

• Change in y-values (rise): -1 - (-7) = 6
• Change in x-values (run): -3 - (-8) = 5
• Slope \(m = \frac{6}{5}\)

This means our line has a slope of \(\frac{6}{5}\), which indicates it rises 6 units for every 5 units it runs to the right.

Point-Slope Form

The point-slope form is a useful tool for writing the equation of a line. It's handy when you have a slope and a specific point on the line. The point-slope equation looks like this:\[ y - y_1 = m(x - x_1) \]You substitute the slope \(m\) and one of the points \(x_1, y_1\) into this formula. Here's how you do it with our example where the slope is \(\frac{6}{5}\) and point (-8, -7):

• Start with the equation: \(y + 7 = \frac{6}{5}(x + 8)\)

This equation still has 'y' and 'x' terms and isn't quite in the standard form yet—but it's a good starting point! The next steps will help transform it into a more conventional form that you might be asked to provide.

Standard Form of a Line

The standard form of a line equation is a tidy way of expressing linear equations. Standard form follows the pattern:\[ Ax + By = C \]where A, B, and C are integers, and A should be a non-negative value. To convert an existing equation to this form, you may need to rearrange the terms and sometimes multiply through by a number to clear out fractions.Continuing with our example in point-slope form \( y + 7 = \frac{6}{5}(x + 8) \), distribute the slope:

• Multiply: \(y + 7 = \frac{6}{5}x + \frac{48}{5}\)
• To eliminate the fraction, multiply everything by 5: \(5y + 35 = 6x + 48\)
• Rearrange to gather like terms: \(6x - 5y = -13\)

This yields a nice clean line equation where all coefficients are integers, giving you a practical way to represent the line passing through the original two points.