Problem 44 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$3 y-5 x=12$$

Step-by-Step Solution

Answer

The slope is \( \frac{5}{3} \).

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by: \[ y = mx + b \] where \(m\) represents the slope and \(b\) the y-intercept. Start by isolating \(y\) on one side of the given equation.

Step 2: Isolate y

Starting with the equation: \[ 3y - 5x = 12 \] First, add \(5x\) to both sides to isolate terms involving \(y\): \[ 3y = 5x + 12 \]

Step 3: Solve for y

Now, divide every term in the equation by 3 to solve for \(y\): \[ y = \frac{5}{3}x + 4 \]

Step 4: Identify the Slope

In the equation \( y = \frac{5}{3}x + 4 \), the coefficient of \(x\) is the slope. Therefore, the slope \(m\) is \( \frac{5}{3} \).

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a simplified way of writing the equation of a line. The general form is given as: \[ y = mx + b \] In this formula, \(y\) represents the dependent variable (usually on the vertical axis), and \(x\) is the independent variable (on the horizontal axis). The term \(m\) denotes the slope of the line, which indicates the steepness or inclination. Lastly, \(b\) stands for the y-intercept, which is the point where the line crosses the y-axis. Here’s a quick rundown of why slope-intercept form is so useful:

• It provides an easy way to identify both the slope and y-intercept.
• Makes graphing a line straightforward.

Whenever you're given a linear equation, converting it to slope-intercept form can help reveal important details about the line.

linear equations

Linear equations represent straight lines when plotted on a graph. A linear equation typically looks like this: \[ax + by = c\] Here, \(a\), \(b\), and \(c\) are constants. The equation can be rearranged into various forms, but the goal is often to isolate \(y\) (or \(x\)) to make the equation easier to understand and graph. Linear equations have various forms:

• Slope-Intercept Form: \( y = mx + b \)
• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
• Standard Form: \( ax + by = c \)

Each of these forms serves different purposes, but slope-intercept is the most commonly used. When converting a linear equation to slope-intercept form, it simplifies the process of determining the slope and y-intercept, which are crucial for graphing the line and understanding its behavior.

isolating variables

Isolating variables is the process of manipulating an equation to get one variable alone on one side, typically as a step towards solving the equation. Let's take a look at how we did this in the given exercise: First, we started with the equation: \[ 3y - 5x = 12 \] To isolate \(y\), we decided to move all terms involving \(x\) to one side. We did this by adding \(5x\) to both sides: \[ 3y = 5x + 12 \] Next, we divided each term by \(3\) to solve for \(y\): \[ y = \frac{5}{3}x + 4 \] Now, \(y\) is isolated on the left side, and we can clearly see the slope (\( \frac{5}{3} \)) and the y-intercept (\( 4 \)). Using the principle of isolating variables makes equations easier to understand and solve. Whether you're working with one-variable or two-variable equations, it's a fundamental skill in algebra. Steps to isolate a variable can always be broken down as:

• Performing operations to move terms around.
• Combining like terms.
• Identifying inverse operations (e.g., adding the opposite, multiplying by the reciprocal).

Understanding this concept simplifies tough-looking problems, making algebra much more manageable.