Problem 77 - Intermediate Algebra

Question

Find the slope of each line. The line with equation \(4 x-3 y=-3\)

Step-by-Step Solution

Answer

The slope of the line is \(\frac{4}{3}\).

Step 1: Understand the Equation

The equation given is in the standard form of a linear equation, which is \(Ax + By = C\). Here, \(A = 4\), \(B = -3\), and \(C = -3\). To find the slope, we need to convert this into slope-intercept form, \(y = mx + b\).

Step 2: Rearrange to Slope-Intercept Form

Start by isolating \(y\) on one side. Begin by subtracting \(4x\) from both sides to get \(-3y = -4x - 3\).

Step 3: Solve for y

Now, divide each term by \(-3\) to solve for \(y\): \(y = \frac{4}{3}x + 1\).

Step 4: Identify the Slope

In the equation \(y = \frac{4}{3}x + 1\), the coefficient of \(x\) represents the slope. Therefore, the slope \(m = \frac{4}{3}\).

Key Concepts

Linear Equation

A linear equation, such as the one we're exploring, represents a straight line on a coordinate plane. These equations are typically written in one of three forms, the most common being the standard form, slope-intercept form, and point-slope form. In this case, we are dealing with an equation in the standard form, which is written as \(Ax + By = C\).
A linear equation includes variables like \(x\) and \(y\), as well as constants like \(A\), \(B\), and \(C\). You're seeing this through the lens of finding the slope. The line will always have a constant rate of change, which is what makes it 'linear'.

• The coefficient of \(x\) changes the steepness or direction of the line.
• The constant \(C\) affects where the line will intersect the axes.

This specific nature makes linear equations a powerful tool to model real-world situations where relationships remain constant over time. A fun fact: linear equations are incredibly simple in concept, yet they play a crucial role in a wide array of mathematical fields.

Slope-Intercept Form

The slope-intercept form of a linear equation is a handy tool when you need to easily find and interpret the slope and y-intercept of a line. This form is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) stands for the y-intercept.
In this form, each element is directly revealing:

• \(m\) indicates how steep the line is. A positive \(m\) means the line inclines as you go from left to right, while a negative \(m\) means it declines.
• \(b\) pinpoints where the line crosses the y-axis, giving you the y-value when \(x = 0\).

This approach to linear equations makes it exceptionally easy to both graph and understand a line's direction and intersection clearly. In our example, after transforming from standard form to slope-intercept form, the line is \(y = \frac{4}{3}x + 1\), showing you that the slope \(m\) is \(\frac{4}{3}\). This line rises 4 units for every 3 it runs horizontally, illustrating how various components of an equation translate directly into graphical features.

Standard Form of a Linear Equation

The standard form is one of the most recognized forms of linear equations, written as \(Ax + By = C\). It's a versatile form and quite useful for quickly determining the general appearance of a line on a graph.
In standard form:

• \(A\) and \(B\) are coefficients that tell us how much the x and y terms contribute to the equation.
• \(C\) is the constant which will affect where the line crosses the axes.

Unlike the slope-intercept form, the standard form doesn’t immediately highlight the slope or y-intercept. To find these values, you’ll often need to rearrange the equation.
The beauty of standard form is in its flexibility. It can be easily converted into slope-intercept or point-slope forms to extract additional information depending on what you're solving for.
The standard form equation \(4x - 3y = -3\) can be converted to \(y = \frac{4}{3}x + 1\), where you can immediately see both the slope and y-intercept, showcasing its adaptability by reformatting the context as needed.