Problem 17 - 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 \(1 \mathrm{~b}\) ) \((-1,-2)\) and \((-6,-7)\)

Step-by-Step Solution

Answer

The equation of the line is \(x - y = 1\).

Step 1: Find the Slope of the Line

To find the slope of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \((-1, -2)\) as \((x_1, y_1)\) and \((-6, -7)\) as \((x_2, y_2)\): \[ m = \frac{-7 - (-2)}{-6 - (-1)} = \frac{-7 + 2}{-6 + 1} = \frac{-5}{-5} = 1 \]. So, the slope \( m \) is 1.

Step 2: Use Point-Slope Form

With the slope \( m = 1 \) and one of the points \((-1, -2)\), we use the point-slope form of the equation \(y - y_1 = m(x - x_1)\). Substitute the values: \[ y - (-2) = 1(x - (-1)) \]. This simplifies to \(y + 2 = x + 1\).

Step 3: Rearrange to Standard Form

Start with the equation from Step 2, \(y + 2 = x + 1\). Subtract \( x \) from both sides to get \( -x + y + 2 = 1 \). Subtract 2 from both sides to isolate the constant on one side: \(-x + y = -1\).

Step 4: Convert to Integer Coefficients

Since \(-1\) is an integer, the expression \(-x + y = -1\) already has integer coefficients. However, the standard form typically starts with a positive \(x\)-term. Multiply the entire equation by \(-1\) to get \(x - y = 1\).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. In algebra, the slope is represented by the letter \( m \). When given two points, the slope tells you how much the line rises or falls as it moves from left to right across a coordinate plane.
To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula helps us calculate the vertical change (rise) divided by the horizontal change (run) between two points. For the points \((-1, -2)\) and \((-6, -7)\), we find the slope as follows:

• Calculate the difference in the \( y \)-coordinates: \(-7 - (-2) = -5\)
• Calculate the difference in the \( x \)-coordinates: \(-6 - (-1) = -5\)
• Divide the differences: \( \frac{-5}{-5} = 1 \)

So the slope \( m \) is 1, indicating the line rises evenly with respect to its run since the slope is positive. Understanding slope is crucial for studying line behavior on graphs.

Point-Slope Form

The point-slope form is an invaluable tool in algebra to describe the equation of a line when a point on the line and its slope are known. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) is the slope, while \((x_1, y_1)\) is a known point on the line.
Using the point \((-1, -2)\) and slope \( m = 1 \), we can form the equation:

• Substitute into the formula: \( y - (-2) = 1(x - (-1)) \)
• Simplify to get: \( y + 2 = x + 1 \)

This form is particularly useful for quickly converting a verbal description of a line into an algebraic representation.
Once you obtain a line's equation in point-slope form, you can easily transition it to other forms, like slope-intercept or standard form, depending on what is required.

Standard Form of a Line

The standard form of a line is a method to express linear equations. Standard form is written as:

• \( Ax + By = C \)

where \( A \), \( B \), and \( C \) are integers, and typically \( A \) should be non-negative.
With the point-slope form equation from earlier, \( y + 2 = x + 1 \), we can convert this to standard form:

• Rearrange terms so that \( x \) and \( y \) are on one side: \( -x + y + 2 = 1 \)
• Simplify to \( -x + y = -1 \)
• Multiply through by \(-1\) to make \( x \) coefficient positive: \( x - y = 1 \)

This rearranged equation \( x - y = 1 \) maintains integer coefficients for \( A \), \( B \), and \( C \). The standard form is particularly useful in applications where integer coefficients provide clearer geometric or real-world interpretations.
It is also important to make sure \( A \) is non-negative, which is why we multiplied the entire equation by \(-1\) to reach the standard form.