Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The line starts at y-intercept (0, 4), then for every 1 unit step to the right on the x-axis, you move down 2 units on the y-axis.

Step 1: Identify the slope and y-intercept

From the equation \(y=-2x+4\), it can be determined that the slope \(m=-2\) and the y-intercept \(c=4\). This means you start at 4 on the y-axis and for every step in x, you move 2 steps down on the y-axis.

Step 2: Plot the y-intercept

Begin by marking the y-intercept (0, 4) on the graph. This represents the point where the line crosses the y-axis.

Step 3: Use the slope to find the next point

The slope is -2. This means for every 1 unit move to the right on the x-axis, you go down 2 units on the y-axis. Starting from the point (0, 4), move 1 unit to the right to 1 on the x-axis, and then move down 2 units to 2 on the y-axis. This gives you the next point (1, 2). Mark this point on the graph.

Step 4: Draw the line

Draw a straight line through the two points. This line represents the graph of the equation \(y=-2x+4\).

Key Concepts

Slope

The slope of a linear equation is a measure of how steep the line is. It shows the rate at which the line ascends or descends as it moves along the x-axis. In the given equation, \(y = -2x + 4\), the slope is represented by -2. This indicates that for every unit you move to the right along the x-axis, the line will drop by 2 units in the y-direction. The slope is denoted as \(m\) and is calculated as

• \( m = \frac{\text{rise}}{\text{run}} \)

Here, the rise refers to the change in the y-values, and the run is the change in the x-values. A negative slope implies a downward trend, moving from left to right.Remember:

• A positive slope means the line rises as it goes from left to right.
• A zero slope implies a horizontal line.
• A vertical line has an undefined slope.

Y-intercept

The y-intercept is the point where the line crosses the y-axis on a Cartesian plane. It is represented by the letter \(c\) in the slope-intercept form of a linear equation which is\(y = mx + c\).In our equation \(y = -2x + 4\), the y-intercept is 4. This means the line touches the y-axis at the coordinate point (0, 4). The y-intercept can be easily found because it is the constant term in the equation when the equation is written in the form of \(y = mx + c\).The y-intercept gives you a fixed point on the graph that you can use to draw the line. Once you pin this point, the slope can guide you to plot another point and graph the line accurately.

Linear Equation

A linear equation is a type of algebraic equation that describes a straight line when plotted on a graph. The standard slope-intercept form of a linear equation is \(y = mx + c\), where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line.
• \(c\) is the y-intercept.

In our case, the equation \(y = -2x + 4\) describes a straight line with a slope of -2 and a y-intercept of 4. A graph of this equation will show how changes in \(x\) affect \(y\), resulting in a straight line tilting downwards from left to right.Linear equations are fundamental in mathematics for modeling and representing relationships between two variables. They are extensively used in various fields like science, engineering, and economics.

Cartesian Plane

The Cartesian plane, also known as a coordinate plane, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). These two axes intersect at a point called the origin, represented by the coordinates (0,0).Points on this plane are identified using ordered pairs \((x, y)\), where:

• \(x\) is the horizontal position from the origin.
• \(y\) is the vertical position from the origin.

In the context of graphing the equation \(y = -2x + 4\), the Cartesian plane provides the framework necessary to plot the line. The y-intercept (0, 4) is located directly on the y-axis, and using the slope, we can determine that moving 1 unit right on the x-axis leads us 2 units down to a new point (1, 2). Graphing on the Cartesian plane makes abstract mathematical concepts tangible by enabling visualization of equations, such as lines, parabolas, and other curves.