Problem 64 - Precalculus with Limits

Question

In Exercises 63 and 64, find the distance between the parallel lines. \(3x - 4y = 1\) \(3x - 4y = 10\)

Step-by-Step Solution

Answer

The distance between the two parallel lines is 1.8 units

Step 1: Identify the Coefficients and Constants from the Line Equations

In the standard form of a line's equation, \(Ax + By = C\), A is the coefficient of x, B is the coefficient of y, and C is a constant. From the given lines: First line: \(A_1 = 3\), \(B_1 = -4\), \(C_1 = 1\), Second line: \(A_2 = 3\), \(B_2 = -4\), \(C_2 = 10\). As we see, \(A_1 = A_2 = 3\) and \(B_1 = B_2 = -4\), we assert that the lines are parallel.

Step 2: Substitute in the Distance Formula

Substitute \(A = 3\), \(B = -4\), \(C_1 = 1\), and \(C_2 = 10\) into the distance formula. This gives us the distance \(d\) as \(d = \frac{|10 - 1|} {\sqrt{3^2 + (-4)^2}}\). Calculate the numerator and denominator separately as it usually makes calculation easier.

Step 3: Calculate the Distance

Evaluate the expression obtained in the previous step. We have \(d = \frac{9} {\sqrt{9 + 16}} = \frac{9} {\sqrt{25}}\). So, the distance \(d\) simplifies to \(d = \frac{9} {5} = 1.8\) units

Key Concepts

Line Equations

Understanding line equations is essential when dealing with the geometry of parallel lines. The standard form of a line equation is expressed as Ax + By = C, where A and B are the coefficients of the variables x and y respectively, and C is a constant term. When we are given two line equations, such as
\(3x - 4y = 1\)
and
\(3x - 4y = 10\),
we observe that the coefficients of x and y are identical in both equations. This signifies that the direction or the 'slope' of these lines are the same, hence confirming that the lines are indeed parallel to each other. To comprehend why this indicates parallelism, imagine walking at the same steepness on two different paths; you would not converge or diverge if those paths had the same steepness, or in mathematical terms, the same slope.

Distance Formula

The distance formula is a vital tool for calculating the shortest distance between two parallel lines. When two lines L1 and L2 are parallel and described by equations in the form Ax + By = C1 and Ax + By = C2 respectively, the distance d between them can be found by the formula:

\[d = \frac{|C2 - C1|} {\sqrt{A^2 + B^2}}\].

In the context of the given exercise, we substitute the respective values of A, B, C1, and C2 into this formula to calculate the distance. The absolute value in the numerator ensures that the distance is always a positive number, while the square root in the denominator represents the Pythagorean Theorem in the context of the gradient of the lines.

Parallel Lines Geometry

The geometry of parallel lines is governed by their defining property: they are always the same distance apart, which means they will never intersect. This consistent separation throughout is what makes calculating their distance possible and essential in various fields of mathematics and applications such as drafting and engineering.

In our exercise, we showed that the lines represented by \(3x - 4y = 1\) and \(3x - 4y = 10\) are parallel by verifying that their coefficients are equal. This parallel property means that if you were to extend both lines infinitely in any direction, they would never meet. Understanding this concept is key to solving real-world problems that involve parallel lines, such as determining the width of a road or railway track, or even finding the optimal placement for certain architectural elements that must run in tandem without converging.