Problem 75 - Algebra for College Students

Question

Let \(A x+B y=C\) and \(A^{\prime} x+B^{\prime} y=C^{\prime}\) represent two lines. Change both of these equations to slopeintercept form, and then verify each of the following properties. (a) If \(\frac{A}{A^{\prime}}=\frac{B}{B^{\prime}} \neq \frac{C}{C^{\prime}}\), then the lines are parallel. (b) If \(A A^{\prime}=-B B^{\prime}\), then the lines are perpendicular.

Step-by-Step Solution

Answer

Equations in slope-intercept form help check that parallel lines have equal slopes and perpendicular lines have slopes with product -1.

Step 1: Convert Equations to Slope-Intercept Form

The general form of a line is given by the equation \( Ax + By = C \). We need to convert this into the slope-intercept form, \( y = mx + b \). For the first line, rearrange to get \( By = -Ax + C \), then divide by \( B \): \( y = -\frac{A}{B}x + \frac{C}{B} \). Similarly, for the second line, \( B^{\prime}y = -A^{\prime}x + C^{\prime} \), divide by \( B^{\prime} \): \( y = -\frac{A^{\prime}}{B^{\prime}}x + \frac{C^{\prime}}{B^{\prime}} \).

Step 2: Verify Parallel Line Condition

To check if the lines are parallel, compare their slopes. For two lines to be parallel, their slopes must be equal. The slopes are \( -\frac{A}{B} \) and \( -\frac{A^{\prime}}{B^{\prime}} \). Equating these slopes, we get \( \frac{A}{A^{\prime}} = \frac{B}{B^{\prime}} \). The lines are parallel if \( \frac{A}{A^{\prime}} = \frac{B}{B^{\prime}} eq \frac{C}{C^{\prime}} \). If this holds true, then the lines do not intersect, confirming they are parallel.

Step 3: Verify Perpendicular Line Condition

For the lines to be perpendicular, their slopes must multiply to \(-1\). This means that \( \left(-\frac{A}{B}\right) \times \left(-\frac{A^{\prime}}{B^{\prime}}\right) = -1 \). Simplifying this condition gives us \( \frac{AA^{\prime}}{BB^{\prime}} = -1 \) or \( AA^{\prime} = -BB^{\prime} \). So, if \( AA^{\prime} = -BB^{\prime} \), the lines are perpendicular.

Key Concepts

Slope-Intercept Form

In mathematics, the slope-intercept form is a way of writing the equation of a line. It's given by the formula \( y = mx + b \). This form is very handy because it instantly tells you two crucial things about the line: the slope \( m \) and the y-intercept \( b \). The slope \( m \) tells us how steep the line is, or how much \( y \) changes for a unit change in \( x \). Meanwhile, \( b \), the y-intercept, shows where the line crosses the y-axis (where \( x = 0 \)).
To convert a standard form equation \( Ax + By = C \) to the slope-intercept form, you solve for \( y \). Start by isolating the term \( By \) on one side: \( By = -Ax + C \). Then, divide each term by \( B \): \( y = -\frac{A}{B}x + \frac{C}{B} \). Voila, you’ve transformed it into the slope-intercept form!

Parallel Lines

Parallel lines are lines in a plane that never meet. They are always the same distance apart. When it comes to their equations, if two lines are parallel, their slopes will be the same.
From the slope-intercept form \( y = mx + b \), the slope is \( m \). So, to determine if two lines are parallel, compare their slopes. If line 1 has the slope \( m_1 = -\frac{A}{B} \) and line 2 has the slope \( m_2 = -\frac{A^{\prime}}{B^{\prime}} \), then the lines are parallel if \( m_1 = m_2 \). This equation simplifies to \( \frac{A}{A^{\prime}} = \frac{B}{B^{\prime}} \).
However, if \( \frac{C}{C^{\prime}} \) is different, it means that while the lines are parallel, they are not the same line (not coincident), thus confirming they never intersect.

Perpendicular Lines

Perpendicular lines intersect to form a perfect right angle (90 degrees) with each other. When it comes to their slopes, the product of their slopes is \(-1\).
In the slope-intercept form, if one line has the slope \(-\frac{A}{B}\) and the other line has the slope \(-\frac{A^{\prime}}{B^{\prime}}\), the lines are perpendicular if \( \left(-\frac{A}{B}\right) \times \left(-\frac{A^{\prime}}{B^{\prime}}\right) = -1 \). This condition simplifies to \( AA^{\prime} = -BB^{\prime} \).
This relationship confirms the orthogonality of two lines in a plane, making sure they meet at right angles.

Slope Formula

The slope of a line is a measure of its steepness and direction. It can be found using the slope formula, which is expressed as \( m = \frac{y_2-y_1}{x_2-x_1} \). The variables \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
The formula tells us how much \( y \) changes for a change in \( x \). A positive slope means the line goes up as you move to the right, whereas a negative slope means it goes down.

• Slope is zero for horizontal lines (constant \( y \)).
• Slope is undefined for vertical lines (constant \( x \)).

Understanding the slope helps us analyze and graph lines, compare line orientations, and solve problems involving linear relationships.