Problem 19 - 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. \((0,0)\) and \((-3,-5)\)

Step-by-Step Solution

Answer

The equation is \(5x - 3y = 0\).

Step 1: Identify the slope formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Use this to find the slope of the line between the points \((0,0)\) and \((-3,-5)\).

Step 2: Calculate the slope

Substitute the points \((0,0)\) and \((-3,-5)\) into the slope formula: \[m = \frac{-5 - 0}{-3 - 0} = \frac{-5}{-3} = \frac{5}{3}.\] Thus, the slope \(m\) is \(\frac{5}{3}\).

Step 3: Use point-slope form

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\). Use point \((0,0)\) and the slope \(m = \frac{5}{3}\) to write the equation: \[y - 0 = \frac{5}{3}(x - 0),\] which simplifies to \(y = \frac{5}{3}x\).

Step 4: Convert to standard form

The standard form of a line is \(Ax + By = C\), where \(A, B,\) and \(C\) are integers. From \(y = \frac{5}{3}x\), multiply through by 3 to eliminate the fraction: \[3y = 5x.\] Rearrange it to \(-5x + 3y = 0\), resulting in \(5x - 3y = 0\).

Key Concepts

Slope Calculation

Understanding how to find the slope is essential in linear equations. The slope measures the steepness or incline of a line connecting two points on a graph. It's represented by the letter \(m\) in equations.
To find the slope between two points, we utilize the slope formula:

• Points: \((x_1, y_1)\) and \((x_2, y_2)\)
• Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

The formula essentially finds the change in \(y\) divided by the change in \(x\). This gives the rise over run.
For example, with points \((0,0)\) and \((-3,-5)\), the slope calculation would be:

• Subtract the \(y\)-values: \(-5 - 0 = -5\)
• Subtract the \(x\)-values: \(-3 - 0 = -3\)

Putting them in the formula gives us \(m = \frac{-5}{-3} = \frac{5}{3}\). Thus, the slope is \(\frac{5}{3}\), indicating for every 3 units moved horizontally, it rises by 5.

Point-Slope Form

The point-slope form is a powerful tool in forming the equation of a line when you have one point on the line and its slope. It is usually expressed as

• \(y - y_1 = m(x - x_1)\)

Having this form allows us to easily substitute the known values and simplify.
Here's how it works in practice:

• Let’s take the point \((0,0)\) and slope \(m = \frac{5}{3}\)
• Substitute these into the point-slope form: \(y - 0 = \frac{5}{3}(x - 0)\)
• Simplify: \(y = \frac{5}{3}x\)

By doing so, we've expressed the line's equation in a simple form where one variable is solved in terms of the other. This is particularly useful for graphing and understanding the line's behavior.

Standard Form of Equation

The standard form of a linear equation is a neat and organized way to display equations, often used in algebra. It appears as:

• \(Ax + By = C\)

The goal is to have integer values for \(A, B,\) and \(C\). This form is preferred in many contexts because it makes handling multiple equations simpler and is easy to read.
Converting from other forms, such as the slope-intercept form, involves a few steps.

• From \(y = \frac{5}{3}x\), multiply through by 3 to remove the fraction.
• This results in \(3y = 5x\).
• Rearrange to get all terms to one side: \(-5x + 3y = 0\)
• Finally, rewrite as \(5x - 3y = 0\) to match the standard form.

This process shows how transformations can be applied to fit the desired equation format, maintaining the integrity and relationship between the variables.