Problem 53 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (6,1)\(;\) parallel to the line \(8 x-y=9\)

Step-by-Step Solution

Answer

The equation of the line is \(8x - y = 47\).

Step 1: Identify the Slope of the Given Line

The given line is in the form \(8x - y = 9\). We need to find its slope in order to write the equation of a parallel line. First, convert the equation to slope-intercept form \(y = mx + b\), where \(m\) is the slope.Rearrange \(8x - y = 9\) to get:\[y = 8x - 9\]So, the slope \(m\) of the given line is 8.

Step 2: Use the Point-Slope Form

Since the lines are parallel, the slope of our desired line will also be 8.We use the point-slope form equation: \[y - y_1 = m(x - x_1)\]Substitute \(m = 8\), \(x_1 = 6\), and \(y_1 = 1\) from the given point (6,1):\[y - 1 = 8(x - 6)\]

Step 3: Simplify the Equation

Now we simplify the equation to get the standard form. First, expand the equation:\[y - 1 = 8x - 48\]Add 1 to both sides:\[y = 8x - 47\]

Step 4: Convert to Standard Form

The standard form of a linear equation is \(Ax + By = C\). To convert, rearrange the equation \(y = 8x - 47\) to:\[8x - y = 47\]Now the equation is in standard form.

Key Concepts

Standard Form

The standard form of a linear equation is an important concept in algebra. This form is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. It is a widely used representation because it can easily handle lines that are vertical or horizontal. In standard form:

• \(A\), \(B\), and \(C\) are integers (not fractions)
• \(A\) should be non-negative (positive or zero)
• The greatest common factor of \(A\), \(B\), and \(C\) should be 1

Using standard form helps in solving systems of linear equations and determining intersections between lines. This form is particularly useful in analyzing equations' coefficients and understanding their geometric meanings. It's a simple yet comprehensive way to present linear equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is very intuitive and expresses the equation terms of its slope and y-intercept. It is written as \(y = mx + b\):

• \(m\) represents the slope of the line
• \(b\) signifies the \(y\)-intercept, where the line crosses the \(y\)-axis

This form allows us to quickly identify the two critical aspects of a line: its steepness and where it intersects the \(y\)-axis. The slope \(m\) indicates how much \(y\) will increase as \(x\) increases by one unit.

The slope-intercept form is typically used for its simplicity and direct visual aid in graphing lines, making it one of the most practical ways to describe linear relationships in a coordinate system.

Parallel Lines

Understanding parallel lines is vital in geometry and algebra. Two lines are considered parallel if they have the same slope but differ in their \(y\)-intercepts. This means that they will never intersect.

• Parallel lines extend in the same direction
• They remain equidistant from each other across the entire length

When given a line in an equation, you can find a parallel line by ensuring the new line equation maintains the same slope as the original. For instance, if a line has the slope \(m = 8\), any parallel line will also have \(m = 8\), no matter where it is positioned in the plane.

In practical terms, parallel lines are useful as they often represent consistent relationships or paths, like railway tracks or patterns.

Point-Slope Form

The point-slope form is an incredibly useful way to write the equation of a line especially when you have a point and a slope. It is expressed as: \(y - y_1 = m(x - x_1)\):

• \(m\) represents the slope of the line
• \((x_1, y_1)\) is a specific point on the line

This form is especially handy when you are given specific coordinates and need to write the equation of the line that runs parallel or perpendicular to a given line.

This method is often a stepping stone to convert to either slope-intercept or standard form, as it directly incorporates both geometrical and functional aspects of the line. It's a flexible tool that aids in understanding how changing coordinates affect the overall equation.