Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=-\frac{3}{4} x+2$$

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{3}{4}x+2\) is a straight line which passes through the points (0,2) and (4,-1).

Step 1: Identify the Slope and Y-intercept

In the given equation \(y=-\frac{3}{4}x+2\), the coefficient of \(x\) is the slope (m) and the constant 2 is the y-intercept (b). So, the slope m is -3/4 and the y-intercept b is 2.

Step 2: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis. This point is always given as (0, b). Since the y-intercept is 2, this point is (0, 2). Plot this point on the y-axis of the graph.

Step 3: Use the Slope to Find Next Point

From the y-intercept, use the slope to get to the next point on the graph. The slope is the change in \(y\) divided by the change in \(x\). Here, the slope is -3/4, which means for every 4 units you move right along the x-axis, you move down 3 units along the y-axis. Starting at (0, 2), move 4 units right to (4, 2), then 3 units down to (4, -1). Plot this point.

Step 4: Draw the Line

Now that we have two points, (0, 2) and (4, -1), we can draw a straight line through these points. This line represents the graph of the equation \(y=-\frac{3}{4}x+2\).

Key Concepts

Slope

The slope of a linear equation is a way of describing how steep a line is. It tells us about the direction and the steepness of a line when graphed. If you encounter a linear equation like the one in this exercise, such as \( y = -\frac{3}{4}x + 2 \), the slope is the number in front of \( x \). In our example, the slope \( m \) is -\( \frac{3}{4} \). This tells us: - The negative sign indicates that the line is going downwards as it moves from left to right across the graph. - The fraction itself means that for every 4 units you move horizontally to the right, the line moves downwards by 3 units.You can think of the slope as "rise over run," where rise is the change in the \( y \)-value (up or down) and run is the change in the \( x \)-value (right or left). It is also a way to predict how changes in one variable (\( x \)) will affect another variable (\( y \)). By understanding the slope, you can easily predict the path of the line on a graph.

Y-Intercept

The y-intercept is another important part of the linear equation. It's the point where the line crosses the y-axis. In a linear equation in the form \( y = mx + b \), \( b \) is the y-intercept.In our example, the equation is \( y = -\frac{3}{4}x + 2 \), so the y-intercept \( b \) is 2. Meaning, the line will intersect the y-axis at the point \( (0, 2) \). This is where the value of \( x \) is zero, and \( y \) becomes 2.The y-intercept gives you a starting point on the graph. Once you plot this initial point, you can use the slope to determine where the line will go next. It helps in creating an accurate representation of the linear equation by rooting the line to a specific point on the graph.

Graphing Linear Equations

Graphing a linear equation means drawing a line that represents all the possible solutions to the equation on the graph. To graph \( y = -\frac{3}{4}x + 2 \), you start by identifying the y-intercept and slope.**Steps for Graphing:**

• Step 1: Identify and plot the y-intercept, which in our case is at point \( (0, 2) \).
• Step 2: Use the slope to locate another point. From the y-intercept, apply the slope \( -\frac{3}{4} \), moving right by 4 units and down by 3 units to reach the new point \( (4, -1) \).
• Step 3: With at least two points found, draw a line through them. This line extends infinitely in both directions and represents the linear equation.

Graphing is a visual representation that helps to understand how variables interact. It is essential for visual learners who find it easier to see the relationships rather than just work with numbers and equations alone.