Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Use slopes to solve Exercises \(39-40\). The line passing through \((-2, y)\) and \((-4,4)\) is perpendicular to the line passing through \((-1,-2)\) and \((4,-1)\) Find \(y\).

Step-by-Step Solution

Answer

\(-6\)

Step 1: Compute the slope of the given line

The coordinates of the two points of the second line are given as \((-1,-2)\) and \((4,-1)\). The slope \(m_2\) of this line can be found using the formula for the slope of a line: \(m_2 = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given values: \(m_2 = ((-1) - (-2)) / (4 - (-1)) = 1/5\)

Step 2: Determine the slope of the unknown line

The unknown line passes through two points: \((-2, y)\) and \((-4,4)\) and it's perpendicular to the given line. The slope of two perpendicular lines are negative reciprocals. Therefore, the slope of the unknown line is the negative reciprocal of the slope of the given line: \(m_1 = - 1 / m_2 = -1 / (1/5) = -5\)

Step 3: Find the value of y

Now we substitute these into the slope formula for the unknown line. This yields the equation \(-5 = (y - 4) / ((-2) - (-4))\). Solving for y gives \(y = -5 * 2 + 4 = -6\)

Key Concepts

Slope Formula

When we want to find the slope of a line that passes through two points, we use the slope formula. The slope is a measure of the steepness of a line, and it is often denoted by the letter "m." To calculate the slope, we take two points on the line, \(x_1, y_1\) and \(x_2, y_2\). The slope formula is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This equation tells us how much the "y" value (vertical change) changes for every one unit of change in the "x" value (horizontal change). It is important to consistently subtract the coordinates in the same order to avoid errors:

• Subtract the y-coordinates to find the change in y.
• Subtract the x-coordinates to find the change in x.

This formula is used whenever you need to find out how a line tilts or inclines in a graph.

Negative Reciprocal

In geometry, lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that if one line has a slope \(m\), then the line perpendicular to it will have a slope of \(-1/m\).
Let's break it down:

• "Reciprocal" means to flip a number. So, the reciprocal of a fraction like \(a/b\) is \(b/a\).
• Adding a "negative" sign changes the sign of the reciprocal. Thus, the negative reciprocal of \(1/5\) is \(-5\).

This concept helps solve problems involving perpendicular lines without plotting them on a graph. Recognizing negative reciprocals is crucial in determining the relationship between two lines.

Solving for Variables

Solving an equation for a variable means finding the value of that variable that makes the equation true. In many cases, this involves using algebraic manipulation. Here's how we solve for the variable "y" when given a slope equation:
In the problem, after finding the negative reciprocal of the slope and using it in our equation for the unknown line, we integrate it into the slope formula. The equation was:
\[ -5 = \frac{y - 4}{(-2) - (-4)} \]
Our goal is to solve for "y." Let's go step-by-step:

• First, simplify the denominator: \((-2) - (-4) = 2\).
• Multiplying both sides by 2 eliminates the fraction: \(-5 \times 2 = y - 4\).
• The equation becomes: \(-10 = y - 4\).
• Add 4 to both sides: \(-10 + 4 = y\).
• Finally, we find \(-6 = y\).

This guides you through step-by-step solutions for finding unknown values in equations.