Problem 13 - Technical Mathematics with Calculus

Question

Write the equation of each straight line and make a graph. Slope \(=4 ; y\) intercept \(=-3\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 4x - 3\). To graph it, plot the y-intercept at \((0, -3)\) and use the slope 4 to find a second point and draw the line.

Step 1: Write the Slope-Intercept Form of the Equation

The slope-intercept form of a linear equation is given by the equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the given slope \(m = 4\) and y-intercept \(b = -3\), the equation of the line can be written by substituting these values into the slope-intercept form.

Step 2: Substitute slope and y-intercept into the Equation

Substitute the given slope \(m = 4\) and y-intercept \(b = -3\) into the slope-intercept form. This yields \(y = 4x - 3\).

Step 3: Graph the Equation

To graph the equation \(y = 4x - 3\), start by plotting the y-intercept \((-3)\) on the y-axis. Then, use the slope to find another point. From the y-intercept, go up 4 units (since the slope is 4) and 1 unit to the right, and plot this second point. Draw a straight line through these two points to represent the graph of the line.

Key Concepts

Understanding Linear Equations

A linear equation is an algebraic equation that forms a line when graphed on a coordinate system. It is the most basic form of an equation used to describe a straight line and typically looks like this:
\( y = mx + b \).
In this equation, \( m \) represents the slope, which indicates the steepness and direction of the line, and \( b \) signifies the y-intercept, which is the point where the line crosses the y-axis.

When you're presented with a linear equation, it tells you all the information you need to graph it. To improve your understanding, remember that any linear equation will give you a straight line graph, and any changes to the values of \( m \) and \( b \) will change the position and angle of the line on the graph. Identifying these components is key to solving and graphing linear equations.

Graphing Straight Lines

The process of graphing straight lines begins by first understanding the format of the linear equation. Once you've identified the slope (\( m \) value) and the y-intercept (\( b \) value), you're ready to start graphing.

First, locate and plot the y-intercept on the vertical axis. This is your starting point. Then, use the slope to determine the rise over run, or in other words, how many units you go up or down for every unit you go right or left.

Pro Tip:

Always move in the direction from left to right as if you're reading a book. This will help you maintain a consistent and correct orientation of the line. For instance, a slope of \( 4 \) means you'll rise 4 units up for every 1 unit you move to the right. Plot the second point based on this movement. Finally, draw a line through your plotted points extending both ends to the edges of your graph, thus creating the graph of your linear equation.

Slope and Y-Intercept

The slope of a line is a measure of its vertical change divided by its horizontal change, commonly expressed as 'rise over run.' It is a crucial concept in understanding how lines behave on a graph. The slope can be positive, negative, zero, or undefined, each indicating a different line orientation:

• Positive slope: line rises to the right
• Negative slope: line falls to the right
• Zero slope: horizontal line
• Undefined slope: vertical line

On the other hand, the y-intercept is simply the location where the line crosses the y-axis. This point is always at \( (0, b) \) because it is where the value of \( x \) is zero. To improve your understanding, consider this: the y-intercept is the starting point of your line on the graph, and the slope tells you how to move from there to draw the entire line. Knowing both of these components allows you to accurately draw any straight line on a graph.