Problem 13 - 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. \((2,3)\) and \((7,10)\)

Step-by-Step Solution

Answer

The equation of the line is \\( 7x - 5y = -1 \\).

Step 1: Find the Slope of the Line

The slope (m) of the line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (2,3) and (7,10), this becomes \( m = \frac{10 - 3}{7 - 2} = \frac{7}{5}. \)

Step 2: Use the Point-Slope Form

With the slope known, use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \). Choose point (2,3): \( y - 3 = \frac{7}{5}(x - 2) \). Simplifying, we have \( y - 3 = \frac{7}{5}x - \frac{14}{5} \).

Step 3: Convert to Slope-Intercept Form

Isolate \( y \) to express the equation in slope-intercept form: \( y = \frac{7}{5}x - \frac{14}{5} + 3 \). Convert 3 to a fraction with a denominator of 5, \( y = \frac{7}{5}x - \frac{14}{5} + \frac{15}{5} \), resulting in \( y = \frac{7}{5}x + \frac{1}{5} \).

Step 4: Clear the Fractions

Multiply the entire equation by 5 to eliminate the fractions: \( 5y = 7x + 1 \).

Step 5: Express in Standard Form

Reorder the equation to match the \( Ax + By = C \) format. Rearrange \( 7x - 5y = -1 \) so all terms are on one side. The equation is already in the correct format.

Key Concepts

Slope Calculation

Finding the slope is essential for understanding how steep a line is between two points. The slope, often represented by the letter \( m \), measures the vertical change relative to the horizontal change. You calculate the slope using the formula:

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

This formula finds how much \( y \) changes when \( x \) changes, often described as "rise over run."
For example, if you have two points, (2,3) and (7,10), you can plug them into the formula:

• \( m = \frac{10 - 3}{7 - 2} = \frac{7}{5} \)

This means that for every 5 units you move horizontally to the right, you'll move up 7 units on the line.

Point-Slope Form

After finding the slope, you can use one of the given points and the slope in the point-slope form of a line equation. This form is useful for writing the initial equation of a line and is expressed as:

• \( y - y_1 = m(x - x_1) \)

For the points given, let's use point (2,3) with the slope \( \frac{7}{5} \):

• \( y - 3 = \frac{7}{5}(x - 2) \)

This equation tells us that the line passes through the point (2,3) and has a slope \( \frac{7}{5} \). Solving it gives us the equation in a simplified form. This form is great for moving to other line equations, like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form makes it easy to identify the slope and the y-intercept at a glance. The formula looks like this:

• \( y = mx + b \)

In this format, \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. Our earlier point-slope form \( y - 3 = \frac{7}{5}(x - 2) \) rearranges into:

• \( y = \frac{7}{5}x + \frac{1}{5} \)

This tells you that the slope is \( \frac{7}{5} \) and it crosses the y-axis just above zero, at \( \frac{1}{5} \). This form is straightforward for graphing and quickly assessing the line's characteristics.

Standard Form of a Line

Sometimes you'll need the line equation in the standard form, \( Ax + By = C \), which is useful in some mathematical contexts like equations involving integers. Starting with our slope-intercept form \( y = \frac{7}{5}x + \frac{1}{5} \), we'll make this conversion.
1. Eliminate the fractions by multiplying the entire equation by 5:

• \( 5y = 7x + 1 \)

2. Rearrange the terms to fit the \( Ax + By = C \) format. Get everything on one side:

• \( 7x - 5y = -1 \)

Now you have the equation in standard form, where A is 7, B is -5, and C is -1. This form can be handy for certain types of algebraic manipulations and provides a neat integer-based equation of the line.