Problem 58 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((-2,-3)\) and is perpendicular to the line \(x+4 y=6\)

Step-by-Step Solution

Answer

The standard form of the equation is \(4x - y = -5\).

Step 1: Determine the slope of the given line

The equation of the given line is \(x + 4y = 6\). Convert this equation into slope-intercept form \(y = mx + b\) to find its slope. Start by isolating \(y\): \(4y = -x + 6\), then divide by 4 to get \(y = -\frac{1}{4}x + \frac{3}{2}\). Hence, the slope of the given line is \(-\frac{1}{4}\).

Step 2: Calculate the perpendicular slope

For a line to be perpendicular to another, its slope should be the negative reciprocal. The given line's slope is \(-\frac{1}{4}\). The negative reciprocal is \(4\). Therefore, the slope of the perpendicular line is \(4\).

Step 3: Use point-slope form to find the equation

Use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is \((-2, -3)\) and \(m = 4\). Substitute the slope and point into the formula: \(y - (-3) = 4(x - (-2))\), simplifying to \(y + 3 = 4(x + 2)\).

Step 4: Simplify to slope-intercept form

Distribute the 4: \(y + 3 = 4x + 8\). Subtract 3 from both sides to isolate \(y\): \(y = 4x + 5\). This is the slope-intercept form of the equation.

Step 5: Convert to standard form

Now convert \(y = 4x + 5\) to standard form \(Ax + By = C\). First, move the \(4x\) to the left side: \(-4x + y = 5\). Multiply the entire equation by \(-1\) to make \(A\) positive: \(4x - y = -5\). The standard form is \(4x - y = -5\).

Key Concepts

Perpendicular Lines

When dealing with lines in geometry and algebra, perpendicular lines are lines that intersect at a right angle, specifically 90 degrees. For two lines to be perpendicular, their slopes must have a special relationship. In mathematical terms, the slopes of two perpendicular lines are negative reciprocals of each other. This means that when you multiply the slope of one line by the slope of another line, the product should always be

• -1

In our given exercise, the line provided is in the form of the equation

• \(x + 4y = 6\)

After simplifying this into the slope-intercept form, the slope (

• m

) is found to be

• \(-\frac{1}{4}\)

To find the slope of a perpendicular line, you must take this slope and find its negative reciprocal, turning \(-\frac{1}{4}\) into

• 4

This new slope indicates the steepness and angle of the line that would intersect the original at a perfect right angle.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is particularly useful because it immediately reveals the slope, which indicates the line's direction and steepness, and the y-intercept, which shows exactly where the line crosses the y-axis. This form is expressed as:

• \(y = mx + b\)

Here,

• \(m\)

is the slope of the line, and

• \(b\)

is the y-intercept. In our exercise, starting with the equation

• \(y = 4x + 5\)

shows us that the perpendicular line has a slope of

• 4

and intersects the y-axis at

• (0,5)

This is derived after using the point-slope form and then simplifying to bring out the slope-intercept form. It serves as a tool to quickly graph the equation or to understand how the line behaves, without additional calculations.

Standard Form

Standard form offers another way to express the equation of a line and is particularly useful for solving systems of equations and understanding linear relationships more cleanly in some contexts. A line's equation in standard form is generally written as:

• \(Ax + By = C\)

In this form,

• A
• B

are coefficients of the variables

• x
• y

and

• C

is a constant. It is important to note that

• A, B,
• and C

are usually integers, and

• A

should be non-negative. In the given problem, the line equation was finally expressed in standard form as

• \(4x - y = -5\)

Converting from slope-intercept form to standard form involves rearranging the terms and often mutiplying through by a constant to eliminate fractions, making it easier to handle in various applications such as analyzing or comparing multiple lines.