Problem 80 - College Algebra Essentials

Question

Find the value of \(y\) if the line through the two given points is to have the indicated slope. (-2, y) \text { and }(4,-4), m=\frac{1}{3}

Step-by-Step Solution

Answer

The value of \( y \) that gives the line a slope of \( \frac{1}{3} \) is -6.

Step 1: Define the slope

We remind ourselves that the definition of the slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are points on the line, and \( m \) is the slope.

Step 2: Insert the given values into the slope formula

We are given the points (-2, y) and (4,-4), along with the slope \( m = \frac{1}{3} \). We substitute these values into the formula : \[ \frac{1}{3} = \frac{-4-y}{4 - (-2)} \]

Step 3: Simplify the equation and solve for y

The equation simplifies to \[ \frac{1}{3} = \frac{-4 - y}{6} \]. Multiply both sides by 6 to isolate \( y \): \[ 2 = -4 - y \] Solving for \( y \) gives \( y = -6 \)

Key Concepts

Understanding the Slope Formula

The slope formula is a foundational concept in coordinate geometry used to determine the steepness of a line and its direction. The formula for the slope (\(m\) ) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the rate at which \(y\) values change as \(x\) values change. A positive slope indicates a line rising from left to right, while a negative slope indicates a line falling from left to right. A slope of zero means the line is horizontal, and an undefined slope (division by zero) implies a vertical line.

In the exercise with points \((-2, y)\) and \((4, -4)\), the slope is given as \(\frac{1}{3}\). This means for every unit increase in \(x\), \(y\) increases by a third of a unit, indicating a gentle uphill slope.

Basics of Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this framework, points on a plane are defined using pairs of numbers. These are usually denoted as \((x, y)\). The \(x\) coordinate represents the horizontal position, and the \(y\) coordinate represents the vertical position.

When analyzing lines in coordinate geometry, key features include slopes, intercepts, and distances. For the exercise, we are interested in finding a particular \(y\) value that will make the slope between two points match a given value. This is done using the slope formula, which connects algebra with geometry by translating spatial changes into numerical terms. By understanding coordinate geometry, we can better visualize how these elements interact on a Cartesian plane.

Algebraic Manipulation Techniques

Algebraic manipulation is a critical skill used to solve equations and find unknown values. In the exercise, we use algebraic manipulation to solve for \(y\) in the slope equation. We start with substituting known values into the slope formula:\[ \frac{1}{3} = \frac{-4-y}{6} \]The next step involves isolating \(y\). We do this by first eliminating the fraction, multiplying both sides by 6, which gives:\[ 2 = -4 - y \].

Next, we solve for \(y\) by adding 4 to both sides, resulting in\[ y = -6 \]. This demonstrates a key algebraic technique: unraveling equations through addition, subtraction, multiplication, and division. These operations help to isolate the variable we are solving for, simplifying the equation step by step. Mastering such techniques makes it easier to work with more complex algebraic expressions in algebra and beyond.