Problem 20 - 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 \(1 \mathrm{~b}\) ) \((5,-8)\) and \((0,0)\)

Step-by-Step Solution

Answer

The equation of the line is \(8x + 5y = 0\).

Step 1: Determine the slope of the line

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \((5, -8)\) and \((0, 0)\), substitute \(x_1 = 5\), \(y_1 = -8\), \(x_2 = 0\), and \(y_2 = 0\):\[m = \frac{0 - (-8)}{0 - 5} = \frac{8}{-5} = -\frac{8}{5}\]

Step 2: Use the slope-point form of the equation

The slope-intercept form of a line is \(y - y_1 = m(x - x_1)\). Use one of the points, say \((0, 0)\), and the slope \(-\frac{8}{5}\):\[y - 0 = -\frac{8}{5}(x - 0)\]Simplify this to get:\[y = -\frac{8}{5}x\]

Step 3: Rearrange into standard form

We need to convert the equation \(y = -\frac{8}{5}x\) to the format \(Ax + By = C\). First, multiply every term by 5 to eliminate the fraction:\[5y = -8x\]Rearrange the terms to get:\[8x + 5y = 0\]

Step 4: Verification

Check that both points \((5, -8)\) and \((0, 0)\) satisfy the equation \(8x + 5y = 0\).1. For \((0,0)\): substituting gives \(8(0) + 5(0) = 0\), which is true.2. For \((5,-8)\): substituting gives \(8(5) + 5(-8) = 40 - 40 = 0\), which is also true.

Key Concepts

Slope Formula

The slope of a line is a fundamental concept in understanding linear equations. It measures the steepness or incline of the line, and can be determined if you know two points on the line. Here's how to find it.
Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula for the slope \( m \) is:

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

Plug in the coordinates of the given points. In our example, these points are \( (5, -8) \) and \( (0, 0) \).
Substitute the values:

• \( m = \frac{0 - (-8)}{0 - 5} = \frac{8}{-5} = -\frac{8}{5} \).

This means for every 5 units moved horizontally to the right, the line moves 8 units down.
Understanding slope helps in visualizing how the line behaves on the graph.

Slope-Intercept Form

The slope-intercept form makes it easy to graph a line and quickly understand its slope and y-intercept. This form is expressed as:

• \( y = mx + b \).

Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
From the previous calculation, we know the slope \( m = -\frac{8}{5} \). Since the line passes through the origin \( (0,0) \), the y-intercept \( b = 0 \). Thus, the slope-intercept form of the equation is:

• \( y = -\frac{8}{5}x \).

This simplified equation allows you to graph the line starting at the origin, using the slope to find other points. Remember, the slope tells you how to move from one point to the next on the graph.
Simply follow the slope: 5 units to the right and 8 units down.

Standard Form of a Line

The standard form of a line is an alternative way to express a linear equation. It is often used because it provides integer coefficients, making it neater and easier to work with, especially in theoretical contexts. The standard form is given by:

• \( Ax + By = C \).

To convert the slope-intercept form \( y = -\frac{8}{5}x \) into standard form, first clear the fraction by multiplying through by 5:

• \( 5y = -8x \).

Now, rearrange the terms to achieve the desired integer format, \( Ax + By = C \):

• \( 8x + 5y = 0 \).

This equation shows that any point \( (x, y) \) lying on the line will satisfy this condition, effectively representing the line in a more universal form.
It makes it easy to perform checks with given points, ensuring they indeed lie on the line.
Remember, the standard form creates a nice, clean perspective of the line with integer values useful for various mathematical applications.