Problem 40 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -3 y=5 x+8 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -5/3, and the y-intercept is -8/3.

Step 1: Rewrite the given equation in slope-intercept form

We are given the equation \(-3y = 5x + 8\). To rewrite it in slope-intercept form, we will isolate y on one side of the equal sign. To do so, first divide both sides of the equation by -3: $$ y = -\frac{5}{3}x - \frac{8}{3} $$ Now, the equation is in slope-intercept form.

Step 2: Identify the slope and y-intercept of the line

From the slope-intercept form of the equation, we can directly read off the slope (m) and y-intercept (b): $$ m = -\frac{5}{3} $$ $$ b = -\frac{8}{3} $$ Therefore, the slope of the line is \(-\frac{5}{3}\), and the y-intercept is \(-\frac{8}{3}\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and appear in various forms. They represent straight lines when graphed on a coordinate plane. A standard way to express a linear equation is through the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) signifies the slope, and \(b\) is the y-intercept of the line.

Linear equations involve constants and variables without exponents, meaning they do not contain squared or higher powers of the variable. When you have an equation consisting of a single variable \(y\) and a term with \(x\), it maintains a direct correlation between these two components.

• Equations in this form make it simple to determine the characteristics of a line, especially its slope and y-intercept.
• Changes in \(x\) directly affect \(y\) linearly, keeping the relationship straightforward.

Understanding linear equations helps in analyzing how data correlates within a set, making them essential in numerous applications such as in economics, biology, and physics.

Slope

The slope of a line in a linear equation is a measure of its steepness. It indicates how much \(y\) changes for a given change in \(x\). In the formula \(y = mx + b\), the \(m\) represents the slope.

The slope is calculated as "rise over run." That means you compare the vertical change (rise) to the horizontal change (run) between two points on the line.

For example:

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right, as seen in the equation \(y = -\frac{5}{3}x - \frac{8}{3}\), with slope \(-\frac{5}{3}\).
• A slope of zero denotes a horizontal line.
• An undefined slope suggests a vertical line.

A steep line is one where the absolute value of the slope is high, indicating a greater vertical change per unit of horizontal movement. Recognizing slopes aids in understanding how two variables are related visually on a graph.

Y-Intercept

The y-intercept is where the line crosses the \(y\)-axis on a graph. In the slope-intercept formula \(y = mx + b\), the \(b\) represents the y-intercept. This value shows where the line starts when \(x = 0\).

For our example equation \(y = -\frac{5}{3}x - \frac{8}{3}\), the y-intercept is \(-\frac{8}{3}\), meaning the starting point of the line on the vertical axis is at \(-\frac{8}{3}\).

Y-intercepts help in:

• Finding the initial value or starting point of the line on a graph.
• Giving insight into the point of contact on the \(y\)-axis, which is often helpful in understanding the initial condition in word problems or real-world scenarios.
• Facilitating the graphing of a line when combined with the slope.

The y-intercept delivers a deeper understanding of the functionality and behavior of a linear equation in practical scenarios, especially when predicting events and trends.