Problem 3 - Elementary Algebra

Question

The following equation are in slope-intercept form. In each case, specify the slope and \(y\) -intercept. $$ y=-5 x-1 ; \quad m=\quad b= $$

Step-by-Step Solution

Answer

Answer: The slope (m) is -5 and the y-intercept (b) is -1.

Step 1: Compare Given Equation to Slope-intercept Form

Compare the given equation, \(y=-5x-1\), with the general slope-intercept form, \(y=mx+b\), to identify the values of m and b.

Step 2: Identify the Slope (m)

The slope, m, is the coefficient of x in the equation. In our given equation, \(y=-5x-1\), the slope m is -5.

Step 3: Identify the y-intercept (b)

The y-intercept, b, is the constant term in the equation. In our given equation, \(y=-5x-1\), the y-intercept b is -1.

Step 4: Write the Final Answer

Now that we have identified the slope and y-intercept, we can write our final answer: The slope, m, is -5 and the y-intercept, b, is -1.

Key Concepts

Understanding the Slope

The concept of slope is fundamental in understanding linear equations, especially those in the slope-intercept form. The slope represents the steepness of a line on a graph. It's calculated as the ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis), often referred to as "rise over run." This is expressed mathematically as: \[ m = \frac{\Delta y}{\Delta x} \] In simpler terms, the slope tells us how much the y value of a point on the line changes when the x value increases by one unit.
For example, if you are skiing down a hill, the slope would tell you how steep the hill is. A steeper hill has a larger slope value. In our equation, \( y = -5x - 1 \), the slope \( m \) is \(-5\).

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• If the slope is zero, the line is horizontal.

Understanding the slope helps in predicting how changes in one variable will affect another.

Exploring the y-intercept

The y-intercept is a key part of the slope-intercept form equation, \( y = mx + b \). It represents the point where the line crosses the y-axis.
When the x-value is zero, the y-intercept is the value of y. In essence, it's where the line starts on the y-axis.In our example equation \( y = -5x - 1 \), the y-intercept \( b \) is \(-1\). This tells us that when \( x = 0 \), \( y = -1 \).
This single point is crucial for drawing the graph, helping to establish the baseline from which the line is drawn.

• The y-intercept provides a starting point for plotting the line on a graph.
• It indicates a particular condition of the line when no changes are yet applied to x.

Grasping the concept of the y-intercept allows you to comprehend the initial conditions of a linear relationship.

Linear Equations in Slope-Intercept Form

Linear equations are expressions of a straight line, typically written in the slope-intercept form: \( y = mx + b \). This form is incredibly useful for quickly determining key characteristics of the line.

• "\( m \)" is the slope, which we have explored, indicating the line's tilt.
• "\( b \)" is the y-intercept, the value where the line meets the y-axis.

Linear equations in this format provide easy insights:

• They allow for immediate recognition of the slope and y-intercept, essential for graphing.
• Simplifying the process of prediction and analysis on how the line behaves on a graph.

In practical terms, understanding linear equations in slope-intercept form empowers you to graph lines swiftly and accurately, and interpret the dynamics of different variables within an equation. Whether you're dealing with data in science or economics, mastering this form illuminates the relationship between two interdependent variables.