Problem 9 - Elementary Algebra

Question

For the following problems, graph the equations. $$ -2 x+y=4 $$

Step-by-Step Solution

Answer

Answer: The slope and y-intercept of the given linear equation are \(m=2\) and \(b=4\). These values can be used to graph the equation by first plotting the y-intercept point (0, 4) on the graph and then using the slope to find another point on the line, such as (1, 6). Finally, connect these points with a straight line to represent the graph of the equation \(-2x+y=4\).

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation \(-2x+y=4\) in slope-intercept form, we need to isolate y: $$ y = 2x + 4 $$ Now, it is in the form \(y = mx + b\), where \(m = 2\) is the slope and \(b = 4\) is the y-intercept.

Step 2: Identify the slope and y-intercept

From the slope-intercept form, we can see that the slope \(m=2\) and the y-intercept \(b=4\). This means that the line passes through the point \((0, 4)\) on the y-axis, and its slope indicates that for every 1 unit increase in the x direction, the line increases 2 units in the y direction.

Step 3: Plot the y-intercept on the graph

Begin by plotting the y-intercept, which is the point \((0, 4)\). Mark this point on the graph.

Step 4: Use the slope to find another point on the line

Since we know that the slope is 2, we can use this information to find another point on the line. From the y-intercept \((0, 4)\), move 1 unit to the right (in the x direction) and 2 units up (in the y direction) to reach the point \((1, 6)\). Mark this point on the graph as well.

Step 5: Draw the line

Now that we have at least two points on the line (\((0, 4)\) and \((1, 6)\)), we can connect them with a straight line. This line represents the graph of the equation \(-2x+y=4\). Here is the final graph for the linear equation \(-2x+y=4\): [Image: Graph of -2x+y=4 with the line passing through points (0,4) and (1,6)]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a straight line. It is written as \(y = mx + b\), where:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept

This form is incredibly useful for graphing because it immediately gives you two key pieces of information: how the line rises or falls, and where the line crosses the y-axis. To convert an equation to this form, solve for \(y\) to isolate it on one side.
In our example, the equation \(-2x + y = 4\) was rewritten as \(y = 2x + 4\). Now it is clear that the slope \(m\) is 2, and the y-intercept \(b\) is 4. This showcases the clean and straightforward nature of the slope-intercept form for quickly identifying graphing elements.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. This point occurs when \(x = 0\), making it particularly easy to identify and plot.
In the slope-intercept form equation \(y = mx + b\), the y-intercept is given by the value \(b\). It tells us exactly where on the y-axis the line meets. This is a cornerstone for starting to graph the line.
For the equation \(y = 2x + 4\), the y-intercept is 4, which means the line crosses the y-axis at point \((0, 4)\). You can easily plot this point on the graph, providing one of the anchor points needed to draw the line accurately.

Plotting Points

Plotting points involves marking specific coordinates on the graph. These points help in forming the line of the equation. Start with the y-intercept as your first point, which you mark directly on the y-axis.

• Begin with the y-intercept, like point \((0, 4)\) in our example.
• Use the slope to find additional points. The slope \(m = 2\) means that for every 1 unit movement to the right (along the x-axis), the line moves 2 units up.

So from \((0, 4)\), you move to \((1, 6)\), allowing another point on the graph. Connecting these points with a straight line gives you the visual representation of the linear equation \(-2x+y=4\). Ensuring accuracy with your points means a precise graph outcome.

Linear Function

A linear function represents a straight line when plotted on a graph. It is characterized by a constant slope, meaning the relationship between \(x\) and \(y\) is consistent. The general formula is \(y = mx + b\).
The words "linear" imply proportionality and straightness.

• "Linear" functions respond predictably with changes in \(x\).
• They manifest as lines without curves, ranging infinitely unless bordered by restrictions.

With the equation from our example, \(y = 2x + 4\), this function is clearly linear because it forms a straight line when graphed. Its slope of 2 indicates that for every increase of 1 in \(x\), \(y\) increases by 2, reinforcing the linearity trait that makes such functions easy to predict and work with.