Problem 19 - 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 \(1 \mathrm{~b}\) ) \((0,0)\) and \((-3,-5)\)

Step-by-Step Solution

Answer

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

Step 1: Identify Slope Formula

To find the equation of the line, we first need to find its slope. The slope (abla) of a line given two points (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Calculate the Slope

Plug the given points (\(0, 0\)) and (\(-3, -5\)) into the slope formula: \( m = \frac{-5 - 0}{-3 - 0} = \frac{-5}{-3} = \frac{5}{3} \). So, the slope of the line is \(\frac{5}{3}\).

Step 3: Use Point-Slope Form

The point-slope form of a line is \( y - y_1 = m(x - x_1) \). Using the point (\(0,0\)) and the slope \(\frac{5}{3}\), the equation becomes \( y - 0 = \frac{5}{3}(x - 0) \), which simplifies to \( y = \frac{5}{3}x \).

Step 4: Convert to Standard Form

To convert the equation \( y = \frac{5}{3}x \) into the standard form \( Ax + By = C \), multiply every term by 3 to eliminate the fraction: \( 3y = 5x \). Rearrange this to get \( 5x - 3y = 0 \).

Step 5: Ensure Coefficients are Integers

In the equation \( 5x - 3y = 0 \), the coefficients are already integers, so this is the final equation in standard form.

Key Concepts

Slope Formula

The Slope Formula is a fundamental concept in geometry and algebra that helps to determine the steepness and direction of a line. To find the slope between two points, we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent two distinct points on a line. By substituting the coordinates of these points into the formula, we can easily calculate the slope \(m\). In our example, with points \((0,0)\) and \((-3,-5)\), we have:- \(y_2 = -5, y_1 = 0\)- \(x_2 = -3, x_1 = 0\)Plugging these values into the formula gives:\[m = \frac{-5 - 0}{-3 - 0} = \frac{-5}{-3}\]This simplifies to \(\frac{5}{3}\). This tells us that for every 3 units moved horizontally, the line rises 5 units vertically.

Point-Slope Form

The Point-Slope Form is an effective way to write the equation of a line when you know the slope and a single point on the line. This format looks like:\[y - y_1 = m(x - x_1)\]Where \(m\) is the slope and \((x_1, y_1)\) is a known point on the line. It's particularly useful because it directly relates the slope and provides an intuitive way to derive further line equations.For instance, using the slope \(\frac{5}{3}\) from our previous section and the point \((0,0)\):- Substitute into the form: \(y - 0 = \frac{5}{3}(x - 0)\)This simplifies to:\[y = \frac{5}{3}x\]This equation represents a line passing through the origin with the slope \(\frac{5}{3}\). It's easy to see the correlation between the slope and the direction of the line from this form.

Standard Form

Standard Form is a popular way to express a linear equation because it arranges the equation into a neat, organized format. The standard form of a linear equation is:\[Ax + By = C\]Where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. This form provides a clear method for identifying the coefficients that relate \(x\) and \(y\).To convert an equation like \(y = \frac{5}{3}x\), first eliminate any fractions by multiplying through by the denominator. This ensures all terms are whole numbers:1. Multiply all terms by 3 to clear the fraction: - \(3y = 5x\)2. Rearrange to bring terms involving \(x\) and \(y\) together on one side: - \(5x - 3y = 0\)This is now in standard form. Here, \(A = 5\), \(B = -3\), and \(C = 0\). It's important to verify that all coefficients are integers to confirm the equation is correctly formatted.