Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$3 x-4 y=12$$

Step-by-Step Solution

Answer

The slope (m) of the line is \(0.75\) and the y-intercept (b) of the line is \(-3\).

Step 1: Isolate y

First, isolate y in the equation \(3x - 4y = 12.\) This can be done by subtracting \(3x\) from both sides of the equation to give \(-4y = -3x + 12\). Then, to further isolate y, you can divide every term in the equation by \(-4\) to give \(y = 0.75x - 3.\)

Step 2: Identify the slope and y-intercept

Now that we have the equation in slope-intercept form (\(y = mx + b\)), we can easily find the slope and the y-intercept. The slope (m) is the coefficient of x, and the y-intercept (b) is the constant term. So, the slope is \(0.75\) and the y-intercept is \(-3\).

Key Concepts

Linear Equations

A linear equation is a type of equation that forms a straight line when graphed on a coordinate plane. It describes a relationship between two variables, usually x and y, and is commonly written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. To solve linear equations, we typically manipulate them to isolate the variables. For example, in the equation \(3x - 4y = 12\), our goal is to express one variable, often y, in terms of the other, x, which is done through algebraic operations such as addition, subtraction, multiplication, and division.
Linear equations are foundational in algebra due to their simplicity and wide applicability. They can represent real-world scenarios such as calculating profit based on sales, predicting outcomes, or even figuring out distances.

• Simple to work with mathematically.
• Forms the backbone for more complex algebraic concepts.
• Easily graphable for visualization.

Slope

The slope of a line is a measure of its steepness and direction. It indicates how much the y-variable changes for every unit change in the x-variable. The slope is determined by the coefficient \(m\) in the slope-intercept form of a linear equation, which is \(y = mx + b\).
In the equation \(y = 0.75x - 3\), the slope is \(0.75\). This tells us that for every unit increase in x, the value of y increases by 0.75 units. The slope is crucial in determining how two variables are related and in predicting future values. A positive slope means the line and relationships rise from left to right, while a negative slope means it falls.

• Positive slope: line goes upward.
• Negative slope: line goes downward.
• Larger slope value indicates a steeper line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. It represents the value of y when x is zero. In the slope-intercept form \(y = mx + b\), the y-intercept is denoted by the constant \(b\).
For the equation \(y = 0.75x - 3\), the y-intercept is at \(-3\). This means the line crosses the y-axis at \(y = -3\). The y-intercept provides a starting point for plotting the graph and understanding the relationship in a practical scenario.

• Visible as the line's intersection with the y-axis.
• Serves as a constant term when x equals zero.
• Provides an initial condition or starting point in applications.