Problem 2 - Elementary Algebra

Question

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(5,2), m=\frac{3}{7}$$

Step-by-Step Solution

Answer

The equation of the line is \(3x - 7y = 1\).

Step 1: Understand the Point-Slope Form of a Line

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. For this exercise, the point is \( (5, 2) \) and the slope \( m \) is \( \frac{3}{7} \).

Step 2: Substitute the Given Values into the Point-Slope Formula

Substitute \( (x_1, y_1) = (5, 2) \) and \( m = \frac{3}{7} \) into the point-slope form, resulting in the equation: \[ y - 2 = \frac{3}{7}(x - 5) \]

Step 3: Simplify the Equation

First, distribute \( \frac{3}{7} \) to both \( x \) and \(-5\): \[ y - 2 = \frac{3}{7}x - \frac{15}{7} \] Add 2 to both sides to combine the constants: \[ y = \frac{3}{7}x - \frac{15}{7} + \frac{14}{7} \] Simplify to: \[ y = \frac{3}{7}x - \frac{1}{7} \]

Step 4: Convert to Standard Form \( Ax + By = C \)

Multiply every term by 7 to eliminate the fractions: \[ 7y = 3x - 1 \] Rearrange to form \( Ax + By = C \): \[ -3x + 7y = -1 \]

Step 5: Ensure Coefficients are Integers

To have positive coefficients, multiply the entire equation by -1 to get: \[ 3x - 7y = 1 \] This is the standard form of the equation you need.

Key Concepts

Point-Slope Form

A very crucial concept in understanding linear equations is the point-slope form. This form is incredibly useful when you know a point on the line and its slope.
In mathematical terms, the point-slope form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a known point on the line, and \(m\) represents the slope.
In our specific exercise, we know that the point on the line is \((5, 2)\) and the slope, \(m\), is \(\frac{3}{7}\).

To find the equation of the line using the point-slope form, you'd substitute these values into the equation:
\[ y - 2 = \frac{3}{7}(x - 5) \]
This equation represents the line in point-slope form. It shows how the line behaves around the given point.
The next step is usually to modify this equation to another common form of a line, particularly the standard form.

Standard Form of a Line

The standard form of a line is a particularly neat way of writing the equation of the line. It's often preferred for its simplicity and ease of interpretation.
The equation in standard form is written as:

• \( Ax + By = C \)

Here, \(A\), \(B\), and \(C\) are integers, and both \(A\) and \(B\) are not simultaneously zero.
In our original exercise, we started from the point-slope form:\[ y = \frac{3}{7}x - \frac{1}{7} \]To convert this into standard form, we aim to eliminate fractions and rearrange terms.
We multiplied through by 7, giving:\[ 7y = 3x - 1 \]Rearranging gives:\[ -3x + 7y = -1 \]To keep coefficients positive, we multiply by \(-1\) getting:\[ 3x - 7y = 1 \]This is the standard form. It makes the line equation easy to interpret visually or graphically.

Slope

The slope of a line is a fundamental concept in graphing linear equations. The slope measures the steepness and direction of a line.
It is usually denoted by \(m\) and can be calculated from two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:

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

The slope determines how the line rises or falls.
A positive slope means the line rises as you move from left to right. A negative slope indicates the line falls. In our exercise, we already knew the slope is \(\frac{3}{7}\), meaning the line rises gently.

This small increase per unit of \(x\) shows a gradual rise across the graph.
Understanding slope is essential because it helps predict the behavior of the line and is integral to forming the line's equation.