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Chapter 2: Problem 69

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The line with equation \(A x+B y+C=0(B \neq 0)\) and the line with equation \(a x+b y+c=0(b \neq 0)\) are parallel if \(A b-a B=0 .\)

Short Answer

Expert verified
The given statement is true. If two lines with equations \(Ax + By + C = 0\) and \(ax + by + c = 0\) have the condition \(Ab - aB = 0\), they are parallel. This is because their slopes, \(m_1 = -\frac{A}{B}\) and \(m_2 = -\frac{a}{b}\), are equal which is the condition for parallel lines.

Step by step solution

01

Remember the formula for slope of a line

The slope of a line given by the equation \(Ax + By + C = 0\), can be calculated by rearranging the equation in the form of \(y = mx + b\) and identifying the value of \(m\). The slope \(m\) can be computed with this formula: $$m = -\frac{A}{B}$$
02

Compute the slopes of given lines

Using the formula from the previous step, compute the slopes of both lines: $$m_1 =-\frac{A}{B}$$ $$m_2 =-\frac{a}{b}$$
03

Verify the parallel condition

Two lines are parallel if their slopes are equal. Therefore, we need to check if \(m_1 = m_2\). $$-\frac{A}{B} = -\frac{a}{b}$$ Cross-multiply: $$Ab = aB$$ Rearrange the equation to match the given condition: $$Ab - aB = 0$$
04

Conclusion

Since the condition \(Ab - aB = 0\) holds true for the equal slopes of two lines, the given line equation is true. If two lines with equations \(Ax + By + C = 0\) and \(ax + by + c = 0\) have the condition \(Ab - aB = 0\), they are parallel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
Understanding the slope of a line is essential when studying coordinate geometry. It is a measure of the steepness or the angle of a line. Mathematically, it is defined as the ratio of the vertical change, or rise, to the horizontal change, or run, between two points on the line.

The general formula to calculate the slope m from a linear equation in the standard form Ax + By + C = 0 is \[m = -\frac{A}{B}\]. This follows from rearranging the equation into the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. The negative sign ensures that the slope is calculated correctly based on the convention of the coordinate plane.

Significance of the Slope

Positive slopes ascend from left to right, while negative slopes descend. A zero slope indicates a horizontal line, and undefined slopes signify vertical lines. By comparing slopes of different lines, their relative steepness, and ultimately, their direction, can become apparent鈥攁n integral part of understanding how lines interact in a plane.
Linear Equations
Linear equations are the foundation of coordinate geometry. They express relationships in which one variable is a linear function of another. The most recognizable form of a linear equation is the slope-intercept form: \[y = mx + b\], where m is the slope and b is the y-intercept. However, they can also be written in standard form Ax + By + C = 0, as seen in our exercise.

The standard form is particularly useful for quickly determining whether two lines are parallel. The coefficients A and B in this form play a pivotal role, since they relate directly to the slope, and consequently, to the condition for parallel lines.

Transforming to Slope-Intercept Form

To find the slope from the standard form, one must isolate y and bring the equation into the slope-intercept form. This transformation underscores the relationship between the slope and the coefficients of the equation, paving the way for a better grasp of how linear equations can mirror geometric properties such as parallelism.
Properties of Parallel Lines
In coordinate geometry, understanding the properties of parallel lines is crucial. Parallel lines are defined as lines in the same plane that never intersect, regardless of how far they are extended on both sides.

For two lines to be parallel in a coordinate plane, they must have the same slope. This is because the slope is a measure of the line's incline, and if two lines have different inclines, eventually, they would meet, which defies the principle of parallelism.

Equation-Based Characterization

Given two lines, if the slope of one line, m1 = -\frac{A}{B}, is equal to the slope of the other line, m2 = -\frac{a}{b}, the lines are parallel. This is expressed algebraically as \[m1 = m2\] or, when applied to our exercise, \[Ab = aB\] which simplifies to \[Ab - aB = 0\].

If the condition Ab - aB = 0 is satisfied, as determined in our step by step solution, it conclusively indicates the two lines are parallel. This property is not only theoretical but can be applied to various real-world contexts such as engineering, design, and architecture, where parallel lines play a critical role.

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