Problem 34 - Algebra A Combined Function

Question

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ -4 x-7 y=9 $$

Step-by-Step Solution

Answer

The slope is \( \frac{4}{7} \).

Step 1: Rearrange the Equation

First, we'll rearrange the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope. The original equation is \( -4x - 7y = 9 \). Start by isolating \( y \).

Step 2: Isolate \( y \)

To isolate \( y \), add \( 4x \) to both sides of the equation: \( -7y = 4x + 9 \).

Step 3: Solve for \( y \)

Now, divide every term by \(-7\) to solve for \( y \): \( y = \frac{4}{7}x + \frac{9}{-7} \).

Step 4: Identify the Slope

In the equation \( y = \frac{4}{7}x - \frac{9}{7} \), the coefficient of \( x \) represents the slope. Thus, the slope \( m \) is \( \frac{4}{7} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a key concept in understanding lines and their behaviors on a graph. It is represented by the equation \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. This form is particularly useful because it allows you to quickly identify the slope and y-intercept from an equation, making it easier to graph the line. Given any linear equation, you can rearrange it to fit this form, providing a clearer picture of the line's direction and position. The slope, \( m \), indicates the steepness and direction of the line:

• If \( m \) is positive, the line rises from left to right.
• If \( m \) is negative, the line falls from left to right.
• A steeper line will have a larger absolute value of \( m \).

By expressing a line in slope-intercept form, you unlock a powerful tool for analyzing and graphing lines.

Linear Equations

Linear equations are equations involving two variables that produce a straight line when graphed. They have the standard form of \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. One of the defining characteristics of a linear equation is that its graph produces a straight line, hence the term 'linear.'The process of working with linear equations often involves converting them into slope-intercept form for ease of interpretation. In their standard form:

• \( A \), \( B \), and \( C \) can be integers or fractions.
• The line will be defined by its slope \( -\frac{A}{B} \) if \( A eq 0 \) and its y-intercept \( \frac{C}{B} \).

Understanding linear equations is crucial in algebra because they are foundational to grasping more complex mathematical concepts. They help in solving many real-world problems, where relationships between quantities are linear. Looking at linear equations as a foundation makes it easier to build upon them with more complex algebraic techniques.

Solving for Y

When given a linear equation, such as \( -4x - 7y = 9 \), isolating \( y \) is an essential step in simplifying and understanding the equation's behavior in graphing. This process involves rearranging the equation to the slope-intercept form, \( y = mx + b \), where \( m \) denotes the slope.To solve for \( y \):

• First, you may need to move the \( x \) term to the other side of the equation using addition or subtraction. For instance, add \( 4x \) to both sides to isolate terms involving \( y \): \( -7y = 4x + 9 \).
• Next, divide the entire equation by the coefficient of \( y \), in this case, \(-7\), to finally express \( y \): \( y = \frac{4}{7}x - \frac{9}{7} \).

Once \( y \) is isolated, the equation is in a clear form showing the slope of the line and how the line interacts with the graph axes. This process helps students not only understand but visualize the relationship between the equation's coefficients and the line it represents.