Problem 30 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(2 x=3 y+6\)

Step-by-Step Solution

Answer

The y-intercept of the line is -2, the x-intercept is 3, and a check point (0,0) is also on the line. This information should allow you to correctly graph the line.

Step 1: Find the Y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Let's substitute \(x = 0\) into \(2 x = 3 y + 6\), it becomes \(0 = 3 y + 6\). Solving for y we get \(y = -2\). Therefore, the y-intercept of the line is -2.

Step 2: Find the X-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Let's substitute \(y = 0\) into \(2 x = 3 y + 6\), it becomes \(2 x = 6\). Solving for x we get \(x = 3\). Therefore, the x-intercept of the line is 3.

Step 3: Choose and Test a Checkpoint

Choosing a checkpoint helps to verify if the line graph is accurate or not. This point should not be one of the intercepts. Generally, \(0,0\) is a good checkpoint to choose if the line does not pass through it. Substituting \(x = 0\) and \(y = 0\) into \(2 x = 3 y + 6\), we get \(0 = 0\). So \(0, 0\) lies on the line.

Step 4: Graph the Equation

Plot the y-intercept (-2) on the y-axis and the x-intercept (3) on the x-axis. Then draw a straight line through these intercepts. The point (0, 0) should also be on the line, verifying that the line was graphed correctly.

Key Concepts

Intercepts

Intercepts are essential points where a line crosses either the x-axis or y-axis on a graph. Knowing these intercepts helps us easily draw the graph of a linear equation.

• Y-intercept: The point where the line crosses the y-axis. It occurs when the variable x is equal to zero.
• X-intercept: The point where the line crosses the x-axis. It happens when the variable y is equal to zero.

Finding these intercepts helps us in plotting the line accurately without having to guess its position on a graph. Simplifying the search and improving the understanding of relationships in algebra.

X-Intercept

To find the x-intercept, we substitute y = 0 in our linear equation and solve for x. This point will tell us where the line crosses the x-axis. For our equation,
\[2x = 3y + 6\] becomes \[2x = 6\] when we set \(y = 0\).
Solving this gives
\[x = \frac{6}{2} = 3\]
Thus, the line crosses the x-axis at the point (3, 0). Plotting this point is a critical step in ensuring our graph is accurate.

Y-Intercept

The y-intercept is found by setting x to zero in your linear equation. This simplifies our problem to only needing to solve for y. Let's illustrate this with our linear equation:
When we set \(x = 0\), the equation \[2x = 3y + 6\] becomes \[0 = 3y + 6\]
Solving for \(y\) results in:
\[3y = -6\]
\[y = -2\]
So the y-intercept is at (0, -2). This point is where the line crosses the y-axis. It’s a fundamental value to accurately draw the line on a graph.

Checkpoint

A checkpoint in graphing serves as a verification step. It ensures that the line you’ve plotted is correct. A good method is to choose a point that isn’t one of the intercepts.
Here's how you can use a checkpoint effectively:

• Select a point, such as (0, 0) if it doesn’t lie on the calculated line.
• Substitute the coordinates into the original equation.
• If the equation holds true, the graph is accurate.

In our worked example, substituting \(x = 0\) and \(y = 0\) into \[2x = 3y + 6\] confirms with \[0 = 0\]. Proving that the point lies on the line.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, finding intercepts and setting up equations represent core skills.
These skills are pivotal as they help us understand and visualize relationships between two variables. Graphing linear equations, like in our example, simplifies complex relationships into an easy-to-read format on a graph.
Through algebra, we can solve for values that outline a line's path across the graph by:

• Substituting values to find intercepts.
• Testing points as checkpoints.

This makes algebra not only about solving equations but analyzing and depicting relationships visually.