Problem 45 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ 5 x+3 y=6 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -5/3 and the y-intercept is 2.

Step 1: Rewrite the equation in slope-intercept form

First, rewrite the given equation to isolate `y`. $$ 5x + 3y = 6 $$ Subtract \(5x\) from both sides of the equation: $$ 3y = -5x + 6 $$ Now divide both sides by 3: $$ y = -\frac{5}{3}x + 2 $$

Step 2: Identify the slope

We can see that the equation is now in slope-intercept form: \(y = mx + b\). The coefficient of \(x\) is the slope, which in this case is \(m = -\frac{5}{3}\).

Step 3: Identify the y-intercept

The constant term in the slope-intercept equation is the y-intercept, which in this case is \(b = 2\).

Step 4: Write the final answer

The slope of the given line is \(m = -\frac{5}{3}\) and the y-intercept is \(b = 2\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is vital for graphing and analyzing lines quickly and effectively. This form is expressed as (y = mx + b), where (m) represents the slope, and (b) represents the y-intercept. The slope is a measure of the steepness and direction of the line, indicating how many units the line rises or falls for each unit it moves horizontally. A positive slope means the line goes upward from left to right, while a negative slope signifies it moves downward.

The y-intercept, on the other hand, is the point where the line crosses the y-axis. It gives you a starting point for drawing your line or understanding where the line would be if extended to cut the y-axis. Converting an equation to this form, as shown in the exercise by isolating (y) and simplifying, allows one to easily extract these two critical pieces of information.

Algebraic Equation

An algebraic equation is a mathematical statement indicating that two expressions are equal, separated by an equals sign. In the example (5x + 3y = 6), variables and constants interact in a way that, if you solve for (y), reveals a particular type of algebraic equation known as a linear equation. By converting this to slope-intercept form, one can understand the specific characteristics of the line represented by the equation.

To solve for (y), algebra involves using operations to isolate the variable on one side of the equation. These can include addition, subtraction, multiplication, division, and sometimes factoring. The goal is to simplify the expression into a format that can be easily interpreted and used, paving the way to graphing and further analysis of the line's properties.

Linear Equations

Linear equations are the simplest and one of the most commonly studied types of algebraic equations. They create straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables, (x) and (y), is (ax + by = c), where (a), (b), and (c) are constants. When (a = 0), the graph is a horizontal line, and when (b = 0), it's a vertical line. In any other case, the graph is an oblique line.

Linear equations are foundational in algebra and they represent many real-world phenomena such as predicting profits, calculating distances, and estimating growth. By mastering operations with linear equations, students gain the tools to interpret and predict change in various contexts, setting a basis for more complex mathematical concepts.