Problem 50 - 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 (8,-3)\(;\) parallel to the line \(6 x+2 y=5\)

Step-by-Step Solution

Answer

Equation: \(3x + y = 21\)

Step 1: Identify the Slope

The given line is in standard form: \(6x + 2y = 5\). First, rewrite this equation in slope-intercept form, \(y = mx + b\), to identify the slope.\[2y = -6x + 5 \]\[y = -3x + \frac{5}{2}\]Thus, the slope \(m\) is \(-3\). Since parallel lines have the same slope, the line we are finding also has a slope of \(-3\).

Step 2: Use the Point-Slope Form

Use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \(m = -3\), \((x_1, y_1) = (8, -3)\). Substitute these values into the equation.\[y + 3 = -3(x - 8)\]

Step 3: Simplify the Equation

Expand the equation from Step 2:\[y + 3 = -3x + 24\]Subtract 3 from both sides to solve for \(y\):\[y = -3x + 21\]

Step 4: Convert to Standard Form

Rearrange the equation \(y = -3x + 21\) to standard form \(Ax + By = C\).Add \(3x\) to both sides:\[3x + y = 21\]Now, the equation is in standard form.

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express a straight line using just two key components: slope and y-intercept. This form is written as:

• \( y = mx + b \)

Here, \( m \) represents the slope, which indicates the steepness or incline of the line.
The variable \( b \) stands for the y-intercept, the point where the line crosses the y-axis.
This form is handy when you need to quickly identify how the line behaves in terms of direction and intersection with the y-axis. Using slope-intercept form is especially useful when comparing different lines or predicting their behavior at various x-values.
In practice, converting a given equation into this format can make solving problems easier and more intuitive. For example, in the provided exercise, transforming the standard equation \(6x + 2y = 5\) to slope-intercept form allows us to easily identify the slope, which is crucial for determining parallel lines.

Identifying and Understanding Parallel Lines

Parallel lines are lines in a plane that never intersect and are always the same distance apart. They have the same slope, but different y-intercepts.
This means no matter how far they are extended, they'll never meet.
To determine if two lines are parallel, you simply compare their slopes.

• If two lines have the same slope \( m \), they are parallel.
• If their slopes are different, they are not.

Given the problem above, the line \(6x + 2y = 5\) when converted to slope-intercept form is \(y = -3x + \frac{5}{2}\).
Thus, the slope is \(-3\). Another line with the same slope \(-3\) would be parallel to this line.
Understanding this concept is essential when tasked with writing equations of lines parallel to a given line, as knowing the slope is the first step to correctly formulating the new line's equation.

Utilizing the Point-Slope Form to Find a Line Equation

The point-slope form is particularly useful when you know the slope of a line and a point it passes through. This form is expressed as:

• \( y - y_1 = m(x - x_1) \)

In this formula, \((x_1, y_1)\) represents the point on the line, and \( m \) is the slope.
The point-slope form is valuable for constructing an equation when given a specific point and a slope, especially when you're working out the equation of a line that is supposed to be parallel to another.To illustrate, using the exercise's specified point \((8, -3)\) and slope \(-3\), we substitute these into the point-slope form, resulting in \( y + 3 = -3(x - 8) \).
This approach not only makes it straightforward to draft the linear equation but also keeps calculations organized as you work towards converting it into other forms, like slope-intercept or standard form. By mastering point-slope form, you gain a powerful tool for tackling various line equation exercises with confidence.