Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Use slopes to solve Exercises \(39-40\). The line passing through \((1, y)\) and \((7,12)\) is parallel to the line joining \((-3,4)\) and \((-5,-2) .\) Find \(y .\)

Step-by-Step Solution

Answer

By following these steps, you will be able to calculate the value of \(y\) which makes the line passing through \((1, y)\) and \((7,12)\) parallel to the line joining \((-3,4)\) and \((-5,-2)\).

Step 1: Calculate the slope of the line joining the points (-3,4) and (-5,-2)

Use the formula for slope which is: \((y_2 - y_1) / (x_2 - x_1)\). Insert for \(x_1 = -3\), \(y_1 = 4\), \(x_2 = -5\) and \(y_2 = -2\) to get the slope.

Step 2: Set up an equation for the slope of the line passing through (1, y) and (7,12) equal to the slope from step 1

The equation for slope is \((y_2 - y_1) / (x_2 - x_1)\). Now set \(x_1 = 1\), \(y_1 = y\) (which is the unknown), \(x_2 = 7\) and \(y_2 = 12\). Set this equation to be equal to the slope calculated from step 1.

Step 3: Solve the equation in step 2 for y

Solving this equation will involve some simple algebra. You should isolate \(y\) on one side of the equation by using algebraic manipulation.

Key Concepts

Slope Calculation

Understanding slope is fundamental when dealing with linear equations and their graphs. The slope of a line measures its steepness and is defined as the rise over run.

Mathematically, it's calculated as the difference in the y-coordinates, commonly referred to as \(y_2 - y_1\), divided by the difference in the x-coordinates, or \(x_2 - x_1\). This formula, \(\frac{y_2 - y_1}{x_2 - x_1}\), embodies the concept of slope by quantifying how much the line moves up or down (\

Parallel Lines

Parallel lines have the same slope and are always the same distance apart, never intersecting. When solving problems involving parallel lines, this property becomes particularly handy since it allows us to set the slopes of two separate lines equal to each other if they’re known to be parallel.

In practical terms, if two lines are parallel, and you know the slope of one, you automatically know the slope of the other. This principle is especially useful in coordinate geometry and can be used to find unknown coordinates, as shown in the original exercise.

Algebraic Manipulation

Algebraic manipulation is the process used to transform an equation or expression into a different form without changing its value. This is often done to isolate a specific variable, making it the subject of the formula—something that's crucial when trying to find the value of an unknown variable.

In the context of the given exercise, algebra is employed to solve for \(y\) by isolating it on one side of the equation. This is achieved through various techniques such as adding or subtracting terms on both sides, multiplying or dividing both sides by a number, and simplifying expressions. When done properly, algebraic manipulation reveals the value of unknown variables, making it an invaluable tool in mathematics.