Problem 11 - Technical Mathematics with Calculus

Question

Find the slope and the \(y\) intercept for each equation, and make a graph. $$y=3 x-5$$

Step-by-Step Solution

Answer

The slope of the equation is 3, and the y-intercept is -5. Plot the point (0, -5), use the slope to find another point, and draw the line.

Step 1: Identify the Slope and Y-intercept

The given equation is in slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given equation \(y = 3x - 5\), the slope (\(m\)) is 3, and the y-intercept (\(b\)) is -5.

Step 2: Plot the Y-intercept

Start by plotting the y-intercept (0, -5) on the graph since the line will cross the y-axis at this point.

Step 3: Use the Slope to Find Another Point

From the y-intercept, use the slope to find another point. Since the slope is 3, which can be written as \(\frac{3}{1}\), move up 3 units and 1 unit to the right from the y-intercept to find the second point (1, -2).

Step 4: Draw the Line

Connect the two points with a straight line to draw the graph of the given equation.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations efficiently. It is represented by the equation \(y = mx + b\), where \(m\) stands for the slope of the line and \(b\) indicates the y-intercept. This form makes it clear and straightforward to identify the steepness and direction of the line (the slope) and the exact point where the line crosses the y-axis (the y-intercept).

When we look at \(y = 3x - 5\), it's apparent that the slope \(m\) is 3 and the y-intercept \(b\) is -5. This means that for every step you move to the right along the x-axis, the value of \(y\) increases by 3 steps, indicating a relatively steep upwards slant to our line.

Y-Intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. It's found when the x-value is zero, thus often expressed as a point with the format \( (0, b) \). In our current equation, the y-intercept is -5, which leads us to the crucial first point to be plotted: \( (0, -5) \).

Importance of the Y-Intercept

Knowing the y-intercept allows us to start plotting the graph of our line with a concrete beginning point. From there, we can apply the slope to determine the rest of the line's course. In real-world scenarios, the y-intercept can be a starting value or an initial condition before any changes represented by the slope begin to take effect.

Plotting Points

Plotting points is the method used for drawing the line on a graph after identifying the slope and y-intercept. Upon plotting the y-intercept, we then use the slope to find the next point. With a slope of 3 or \(\frac{3}{1}\), from the y-intercept at \( (0, -5) \), we move up 3 units (because the slope is positive) and 1 unit to the right, guiding us to the next point at \( (1, -2) \).

Creating the Line

After identifying at least two points through the slope-intercept approach, we simply connect these dots with a straight line. This visual representation helps to unlock a better understanding of the relationship between the variables in the equation. Furthermore, plotting multiple points can ensure the accuracy of the graph.