Problem 14 - 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. \((1,4)\) and \((9,10)\)

Step-by-Step Solution

Answer

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

Step 1: Find the Slope

First, determine the slope of the line using the formula for the slope between two points, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points \((1,4)\) and \((9,10)\), substitute the coordinates into the formula to get \( m = \frac{10 - 4}{9 - 1} = \frac{6}{8} = \frac{3}{4} \).

Step 2: Use Point-Slope Form

With the slope calculated, use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \). Selecting the point \((1,4)\), the equation becomes \( y - 4 = \frac{3}{4}(x - 1) \).

Step 3: Simplify the Equation

Expand the equation from Step 2: \( y - 4 = \frac{3}{4}x - \frac{3}{4} \). Add 4 to both sides to get \( y = \frac{3}{4}x + \frac{13}{4} \).

Step 4: Eliminate Fractions

Multiply every term by 4 to eliminate the fractions: \( 4y = 3x + 13 \).

Step 5: Rearrange to Standard Form

Rearrange the equation into the standard form \( Ax + By = C \). Move 3x to the left side to get \( -3x + 4y = 13 \). Multiply through by -1 to ensure A is positive, resulting in the final form: \( 3x - 4y = -13 \).

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. It is calculated between two points on the line. The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \( m \) is the slope
• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points

This formula measures the vertical change (rise) over the horizontal change (run).
For example, if we use the points \((1, 4)\) and \((9, 10)\), then:\[ m = \frac{10 - 4}{9 - 1} = \frac{6}{8} = \frac{3}{4} \]This means the line increases by 3 units vertically for every 4 units it moves horizontally.

Point-Slope Form

Once you have the slope, you can use the point-slope form of a line to create an equation. This form is particularly useful for writing the equation when you know a point on the line and its slope. The point-slope form is given by:\[ y - y_1 = m(x - x_1) \]This equation allows you to easily plug in the slope and any point \((x_1,y_1)\) from the line.
Using our slope \( \frac{3}{4} \) and point \((1,4)\), the equation becomes:\[ y - 4 = \frac{3}{4}(x - 1) \]Here, the form clearly shows how alterations in the \( x \) value will affect \( y \). This form is an excellent bridge to rearranging the equation into different forms, as seen in later steps.

Standard Form of Linear Equation

The standard form of a linear equation is expressed as \[ Ax + By = C \]where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative value.
To convert our current equation from point-slope form to standard form, we rearrange terms as follows:
Starting from \[ y = \frac{3}{4}x + \frac{13}{4} \]multiply all terms by 4 to eliminate fractions, leading to:\[ 4y = 3x + 13 \]Next, rearrange this as:\[-3x + 4y = 13 \]Finally, multiply the entire equation by -1 to make \( A \) positive:\[ 3x - 4y = -13 \]This is now the standard form of the original line equation.

Elimination of Fractions

Eliminating fractions from equations is an essential step in simplifying expressions, especially when expressing them in standard form.
Fractions can make equations harder to understand and manipulate. Hence, multiplying through by the denominator is a helpful strategy.

• First, identify the least common denominator in the equation.
• Multiply every term by this denominator to clear the fractions.

Consider the equation \[ y = \frac{3}{4}x + \frac{13}{4} \]By multiplying through by 4, you eliminate the fractions:\[ 4y = 3x + 13 \]This step is crucial before rearranging terms to get the equation into standard form. It ensures simpler and more manageable calculations in further transformations.