Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$2 x+y=4$$

Step-by-Step Solution

Answer

The graph of the equation \(2x + y = 4\) is a straight line passing through points (2,0), (0,4), and (1,2).

Step 1: Finding the X-intercept

First, to find the x-intercept, set \(y=0\) in the equation and solve for \(x\). This gives: \(2x + 0 = 4\), which simplifies to \(2x = 4\). Solving for \(x\) gives \(x = 2\). So the x-intercept is at (2,0).

Step 2: Finding the Y-intercept

Similarly, set \(x=0\) to find the y-intercept. The equation becomes \(2*0 + y = 4\), which simplifies to \(y = 4\). So the y-intercept is at (0,4).

Step 3: Choosing and Testing a Checkpoint

The point (1,2) is neither the x-intercept nor the y-intercept, so it's a good candidate for a checkpoint. Substitute \(x = 1\) and \(y = 2\) into the equation \(2x + y = 4\). This yields \(2*1 + 2 = 4\), which is true, so the point lies on the line.

Step 4: Drawing the graph

Finally, draw the graph. Plot the points (2,0), (0,4), and the checkpoint (1,2). Draw a line through those points; this is the graph of the equation.

Key Concepts

Intercepts

When learning how to graph linear equations, intercepts are key points where the line crosses the axes on the coordinate plane.
The intercepts can help you quickly sketch the line without needing to plot multiple points.

• X-intercept: This is the point where the line crosses the x-axis. At this point, the value of \( y \) is zero. To find it, set \( y = 0 \) in your equation and solve for \( x \).
• Y-intercept: This is the point where the line crosses the y-axis, meaning \( x = 0 \). To find it, set \( x = 0 \) in your equation and solve for \( y \).

For example, in the equation \( 2x + y = 4 \), setting \( y = 0 \) gives us the x-intercept \( (2, 0) \), and setting \( x = 0 \) gives us the y-intercept \( (0, 4) \).
These intercepts provide two important reference points to begin sketching your line.

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by two axes: the horizontal x-axis and the vertical y-axis.
This plane allows us to graph equations and visualize solutions using coordinates. Every point on this plane is defined by a pair of values, \( (x, y) \).

Understanding the coordinate plane is essential when graphing linear equations.

• Quadrants: The plane is divided into four sections, known as quadrants. Each quadrant has a unique combination of positive and negative values for \( x \) and \( y \).
• Origin: The point \( (0, 0) \) where the two axes intersect is known as the origin.

By plotting intercepts and other key points onto this plane, you can draw accurate graphs of equations like \( 2x + y = 4 \).
This visualization helps in understanding the linear relationships represented by the equation.

Equation of a Line

An equation of a line in a two-dimensional space relates the x and y coordinates for any point on the line.
Commonly, the equation takes the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

The equation not only defines the line but also provides a mathematical way to describe it.
Here’s how to understand it:

• Slope: This depicts the steepness or direction of the line. In the formula \( y = mx + c \), \( m \) represents the slope.
• Intercepts: From the standard form \( ax + by = c \), intercepts can be directly calculated to assist in graphing.

For example, \( 2x + y = 4 \) can be manipulated to show its slope and y-intercept in slope-intercept form. Understanding the equation helps in predicting and drawing the line on the graph.

Checking Solutions

Once the graph of an equation is drawn using intercepts and the coordinate plane, it's essential to check that the graphical solutions fulfill the equation.
This involves verifying whether certain points, also known as checkpoints, actually lie on the line defined by the equation.

• Selecting a Checkpoint: Choose a point not already used to find intercepts. It should be simple to calculate.
• Substitute and Confirm: Insert the x and y values of your checkpoint into the equation. If both sides of the equation balance, the point is part of the line.

For instance, using the point \( (1, 2) \) for the equation \( 2x + y = 4 \), substitute \( x = 1 \) and \( y = 2 \).
You verify the calculation \( 2(1) + 2 = 4 \) which confirms this point lies on the line.
This step is crucial for ensuring your graph accurately reflects the equation and assists in identifying any potential mapping errors.