Problem 14 - 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 \(1 \mathrm{~b}\) ) \((1,4)\) and \((9,10)\)

Step-by-Step Solution

Answer

The equation is \(3x - 4y = -13\).

Step 1: Calculate the Slope

To find the slope \(m\) of the line passing through the points \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (9, 10)\), use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 4}{9 - 1} = \frac{6}{8} = \frac{3}{4}.\]Hence, the slope \(m\) is \(\frac{3}{4}\).

Step 2: Use Point-Slope Form

With the slope \(m = \frac{3}{4}\) and one of the points, \((1, 4)\), use the point-slope formula for a line: \[y - y_1 = m(x - x_1).\]Substitute the values:\[y - 4 = \frac{3}{4}(x - 1).\]

Step 3: Simplify to Slope-Intercept Form

Expand and simplify the equation from Step 2:\[y - 4 = \frac{3}{4}x - \frac{3}{4}.\]Add 4 to both sides to solve for \(y\):\[y = \frac{3}{4}x - \frac{3}{4} + 4.\]Convert 4 to a fraction:\[y = \frac{3}{4}x + \frac{16}{4} - \frac{3}{4} = \frac{3}{4}x + \frac{13}{4}.\]

Step 4: Convert to Standard Form

Multiply every term by 4 to eliminate the fractions:\[4y = 3x + 13.\]Rearrange to put in standard form \(Ax + By = C\):\[3x - 4y = -13.\]

Key Concepts

Slope

The slope is a measure of how steep a line is. It defines the relationship between changes in the vertical direction to changes in the horizontal direction. To find the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:

• \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

This equation calculates how much the \(y\)-coordinates change for a unit change in the \(x\)-coordinates. Simply put, the slope tells you how much "rise" (vertical change) corresponds to a "run" (horizontal change).
In our exercise, two points \((1, 4)\) and \((9, 10)\) were given. Applying the slope formula, we see:

• \[m = \frac{10 - 4}{9 - 1} = \frac{6}{8} = \frac{3}{4}\]

So, the slope of the line is \(\frac{3}{4}\), which means for every increase of 4 in the horizontal direction, there is an increase of 3 in the vertical direction.

Point-Slope Form

The point-slope form is particularly useful when you know a single point on the line and the slope. The general formula is:

• \[y - y_1 = m(x - x_1)\]

where \(m\) represents the slope and \((x_1, y_1)\) is any point on the line.
Let’s see how this applies to our exercise using the point \((1,4)\) and the slope \(\frac{3}{4}\):

• \[y - 4 = \frac{3}{4}(x - 1)\]

This setup allows you to easily substitute any values and solve for the equation of the line. It directly links the coordinate change to the predefined slope, making it easier to visualize how slopes work.
The point-slope form is great for quickly transferring information directly from known values to an equation format without conversion right away, enabling a smooth transition to other forms if desired.

Standard Form

The standard form of a line's equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. This form is often preferred for its organization because both variables, \(x\) and \(y\), are on one side of the equation.
To convert an equation from slope-intercept or point-slope form to standard form, it is usually necessary to eliminate fractions, rearrange terms, and ensure that the coefficients are integers.In our exercise, after simplifying the equation derived from point-slope form:

• \[y = \frac{3}{4}x + \frac{13}{4}\]

We multiply every term by 4 to clear the denominators:

• \[4y = 3x + 13\]

Then rearrange the terms to get:

• \[3x - 4y = -13\]

In standard form, this equation \(3x - 4y = -13\) clearly represents a complete integer-based formula for the line. This format is often used in algebra to provide a neat, standard reference.