Problem 46 - Intermediate Algebra

Question

Contains the point \((-2,-3)\) and is perpendicular to the line \(x+4 y=6\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 4x + 5 \).

Step 1: Identify the Slope of the Given Line

The equation of the original line is given as \( x + 4y = 6 \). We need to rewrite this in slope-intercept form \( y = mx + b \), where \( m \) is the slope. First, solve for \( y \): \[ 4y = -x + 6 \] \[ y = -\frac{1}{4}x + \frac{6}{4} \] This implies the slope \( m = -\frac{1}{4} \).

Step 2: Determine the Slope of the Perpendicular Line

Lines that are perpendicular to each other have slopes that are negative reciprocals. Thus, if the slope of the original line is \(-\frac{1}{4}\), the slope of the line perpendicular to it is \( 4 \) (the negative reciprocal of \(-\frac{1}{4}\)).

Step 3: Use Point-Slope Form to Write the New Equation

Use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point that the line passes through, and \( m \) is the slope. Using the point \((-2, -3)\) and slope \( m = 4\), substitute them into the formula:\[ y - (-3) = 4(x - (-2)) \] which simplifies to \[ y + 3 = 4(x + 2) \].

Step 4: Simplify the Equation

Now, expand and simplify:\[ y + 3 = 4x + 8 \] Subtract 3 from both sides to solve for \( y \):\[ y = 4x + 5 \] This is the equation of the line that contains the point \((-2, -3)\) and is perpendicular to the line \( x + 4y = 6 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most straightforward ways to express a linear equation. In this form, the equation of a line looks like this: \[ y = mx + b \]. Here, \( m \) represents the slope of the line and \( b \) is the y-intercept, which is where the line crosses the y-axis.
To find the slope-intercept form from a standard form equation like \( x + 4y = 6 \), you first need to solve for \( y \). This involves manipulating the equation to isolate \( y \) on one side, as seen in the steps:

• Move \( x \) to the other side: \( 4y = -x + 6 \)
• Divide everything by 4: \( y = -\frac{1}{4}x + \frac{6}{4} \)

This final equation, \( y = -\frac{1}{4}x + \frac{3}{2} \), gives us a slope \( m = -\frac{1}{4} \) and y-intercept \( b = \frac{3}{2} \).
Knowing how to convert an equation into slope-intercept form is a basic but crucial skill when dealing with lines, as it allows you to quickly identify the slope and y-intercept.

Point-Slope Form

Point-slope form is a helpful tool when you have a point on a line and the slope but not necessarily the y-intercept. The point-slope form is expressed as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) is the slope.
In our exercise, we are asked to find the equation of a line with a specific slope passing through a certain point \((-2, -3)\). The calculated slope from the negative reciprocal concept is \(4\). Substituting these into the point-slope formula, we get:

• \( y - (-3) = 4(x - (-2)) \)
• This simplifies to \( y + 3 = 4(x + 2) \)

You can see how straightforward it is to write the line equation using point-slope form. Later, you can easily convert it into slope-intercept form, making it flexible in use for various algebraic scenarios.

Negative Reciprocals

The concept of negative reciprocals is key to understanding perpendicular lines. If you have two lines that are perpendicular, their slopes are negative reciprocals of each other. This means if one line has a slope of, say \(-\frac{1}{4}\), the slope of the line perpendicular to it would be \(4\).
Calculating the negative reciprocal involves two steps:

• Find the reciprocal: Flip the slope fraction. For \(-\frac{1}{4}\), the reciprocal is \(4\).
• Change the sign: Invert the sign, so \(-\frac{1}{4}\) changes to a positive \(4\).

Understanding negative reciprocals is vital as it allows us to determine the slope of a perpendicular line effortlessly, which is often required in both algebra and geometry.

Line Equation

Finding the equation of a line gives you a complete picture of how the line behaves across a plane. Generally, there are different forms you might encounter or need to convert to, such as slope-intercept form and point-slope form.
In solving for the equation of the line in our exercise, we initially used the point-slope form to incorporate the given point \((-2, -3)\) and calculated slope \(4\). Afterward, it was simplified to a more familiar slope-intercept form to yield the final line equation:

• From \( y + 3 = 4(x + 2) \), simplify to \( y = 4x + 8 - 3 \)
• Resulting equation: \( y = 4x + 5 \)

Understanding the full process of deriving a line equation from different forms ensures that you'll be capable of tackling various algebra problems confidently. Whether you're given a point, a slope, or needing to find perpendicular lines, line equations are fundamental to representing these relationships mathematically.