Problem 39 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ 5 y+4 x=6 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -\(\frac{4}{5}\) and the y-intercept is \(\frac{6}{5}\).

Step 1: Write the given equation in the standard form

To convert the equation \(5y + 4x = 6\) to the standard form (\(y = mx + b\)), we need to isolate the y variable. Do this by following these steps: 1. Subtract 4x from both sides of the equation. 2. Divide by 5 on both sides. The transformed equation will look like this: $$ y = -\frac{4}{5}x + \frac{6}{5} $$

Step 2: Identify the slope (m) and y-intercept (b) from the equation

Now that the equation is in the standard form, we can easily identify our slope and y-intercept: 1. The slope (m) is the coefficient in front of the x term. In this case, m = -\(\frac{4}{5}\). 2. The y-intercept (b) is the constant term in the equation. In this case, b = \(\frac{6}{5}\).

Step 3: Write the final answer

After identifying slope (m) and y-intercept (b) from the equation, we can write the final answer as: slope (m) = -\(\frac{4}{5}\), y-intercept (b) = \(\frac{6}{5}\).

Key Concepts

Understanding the Slope

The slope of a line in a linear equation represents the degree of steepness and the direction in which the line climbs or descends. Mathematically, it is expressed as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run). A positive slope means the line ascends from left to right, while a negative slope indicates the opposite. For instance, in the standard form of the equation

• \( 5y + 4x = 6 \) converted to \( y = mx + b \), the slope \( m \) is \(-\frac{4}{5}\).

This indicates the line descends slightly as it moves from left to right as the change in y is less pronounced compared to x. By understanding the slope, one can graph the tilt and alignment of a line accurately.

Deciphering the Y-Intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. It tells us the value of y when x is zero. In real-world situations, it signifies the starting point or initial value before any changes happen due to x.

• For the transformed equation \( y = -\frac{4}{5}x + \frac{6}{5} \), the y-intercept is \( \frac{6}{5} \) or 1.2.

This gives a concrete point on the graph, showing where the line strikes the y-axis. By locating this point graphically, you can accurately predict the line’s behavior even if other data points are not apparent.

Delving into Linear Equations

Linear equations are fundamental equations consisting of two variables: x and y, and always produce a straight line when graphed. They are usually expressed in the form \( y = mx + b \), known as the slope-intercept form, where:

• \( m \) is the slope,
• \( b \) is the y-intercept.

For the exercise, we started with an equation \( 5y + 4x = 6 \) and transformed it into this form to identify important components like slope and y-intercept. Understanding linear equations help visualize relationships between variables, make predictions, and solve real-life problems efficiently.

Insights into the Standard Form of Equations

The standard form of a linear equation is typically expressed as \( Ax + By = C \). This format is especially useful because it establishes clear coefficients and constants that can provide a clear understanding of the relationship between x and y terms. In standard form:

• \( A \), \( B \), and \( C \) are integers,
• and \( A \) should be a positive number.

However, when it comes to easily identifying the slope and y-intercept, converting these equations into the slope-intercept form \( y = mx + b \) as shown in the exercise with \( 5y + 4x = 6 \) is preferred. This conversion simplifies the process of understanding the line’s visual representation and analytic simplicity.