Problem 54 - Elementary Algebra

Question

Contains the point \((-1,4)\) and is parallel to the line \(x-2 y=6\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{2}x + \frac{9}{2}\).

Step 1: Understanding the Problem

We need to find the equation of a line that passes through the point \((-1, 4)\) and is parallel to the line given by the equation \(x - 2y = 6\). A line parallel to another line has the same slope.

Step 2: Find the Slope of the Given Line

Convert the given line equation \(x - 2y = 6\) into slope-intercept form \(y = mx + b\). Start by subtracting \(x\) from both sides: \(-2y = -x + 6\). Divide the equation by \(-2\) to solve for \(y\): \[ y = \frac{1}{2}x - 3 \]The slope \(m\) of the given line is \(\frac{1}{2}\).

Step 3: Use the Point-Slope Form

Since our line is parallel to the given line, it will have the same slope \(m = \frac{1}{2}\). Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through, which is \((-1, 4)\).Substitute \(m = \frac{1}{2}\), \(x_1 = -1\), and \(y_1 = 4\) into the formula: \[ y - 4 = \frac{1}{2}(x + 1) \]

Step 4: Simplify the Equation

Distribute \(\frac{1}{2}\) on the right side: \[ y - 4 = \frac{1}{2}x + \frac{1}{2} \] Add \(4\) to both sides to solve for \(y\): \[ y = \frac{1}{2}x + \frac{1}{2} + 4 \]Combine the constant terms: \[ y = \frac{1}{2}x + \frac{9}{2} \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is crucial for understanding the underlying structure of linear equations. It's a way to quickly determine both the slope and the y-intercept of a line. This form of a line equation is typically written as \(y = mx + b\), where:

• \(m\) denotes the slope of the line, indicating how much \(y\) changes for a change in \(x\).
• \(b\) represents the y-intercept, which is where the line crosses the y-axis.

To convert any linear equation into this form, simply solve it for \(y\). For instance, starting with our example \(x - 2y = 6\), we worked our way to \(y = \frac{1}{2}x - 3\) to identify that the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(-3\). This conversion helps in quickly evaluating the behavior and position of the line on a graph.

Point-Slope Form

Understanding the point-slope form is vital especially when we know a specific point on the line and its slope. This convenient form is expressed as \(y - y_1 = m(x - x_1)\), where:

• \(m\) is the slope of the line.
• \((x_1, y_1)\) is a known point the line passes through.

This formula is particularly useful in situations where you have a point and the slope, rather than the full line equation. In our exercise, we found the slope \(m = \frac{1}{2}\), and used \((-1, 4)\) as our point. By substituting these values into the point-slope form, we arrived at \(y - 4 = \frac{1}{2}(x + 1)\). This form highlights how the line is strategically positioned around the point \((-1, 4)\), and from here, it can be further simplified.

Line Equation

A line equation represents the relationship between the x and y coordinates on a 2-dimensional plane. The most common forms are the slope-intercept form and the point-slope form. The process of deriving a specific line equation depends largely on the information given:

• If you have the slope and the y-intercept, you use the slope-intercept form \(y = mx + b\).
• If you have a point and a slope, go for the point-slope form \(y - y_1 = m(x - x_1)\).

Each form can be used to establish an equation easily. In our original exercise, we started with an equation of a line that needed to be parallel to another – meaning they share the same slope. Given the slope \(\frac{1}{2}\) from one line's equation and a point \((-1, 4)\), we crafted a new line's equation: \(y = \frac{1}{2}x + \frac{9}{2}\). This new equation is ready to describe the line's path through its coordinate plane clearly, helping you understand its behavior and directionality.