Problem 44 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(3 x+2 y=6\)

Step-by-Step Solution

Answer

The x-intercept is \((2, 0)\) and the y-intercept is \((0, 3)\). Graph the line through these points.

Step 1: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 3x + 2y = 6 \). This simplifies to \( 3x = 6 \). Dividing both sides by 3 gives us \( x = 2 \). Therefore, the x-intercept is \((2, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 3x + 2y = 6 \). This simplifies to \( 2y = 6 \). Dividing both sides by 2 gives us \( y = 3 \). Therefore, the y-intercept is \((0, 3)\).

Step 3: Graph the Equation

To graph the equation, plot the x-intercept \((2, 0)\) and the y-intercept \((0, 3)\) on the coordinate plane. Draw a straight line through these two points. This line represents the graph of the equation \(3x + 2y = 6\).

Key Concepts

Understanding the x-intercept

The x-intercept of a graph is a fundamental concept in linear equations. It represents the point where the graph of the equation crosses the x-axis. At this point, the y-value is zero.
To find the x-intercept, follow these steps:

• Set the y-variable to zero in the equation. This is because, at the x-intercept, the graph is touching the x-axis, meaning there's no vertical displacement (y = 0).
• Solve the resulting equation for x.

In our example equation, \(3x + 2y = 6\), setting \(y = 0\) gives \(3x = 6\). By dividing both sides by 3, we find \(x = 2\). Thus, the x-intercept is at point \((2, 0)\). This simple process allows us to pinpoint where the graph meets the x-axis. Recognizing intercepts helps in quickly sketching and understanding the behavior of linear graphs.

Understanding the y-intercept

The y-intercept is equally crucial in linear equations, representing the point where the graph crosses the y-axis. At this point, the x-value is zero, as the graph touches the y-axis directly. Finding the y-intercept involves:

• Setting the x-variable to zero in the equation. Since the x-coordinate is zero at this point, the graph is crossing the y-axis.
• Solving the simplified equation for y.

In the example equation \(3x + 2y = 6\), we set \(x = 0\), resulting in \(2y = 6\). Dividing by 2, we find \(y = 3\). Therefore, the y-intercept is at \((0, 3)\). The knowledge of y-intercepts helps visualize where a line will intersect the y-axis, which is essential in plotting and interpreting graphs.

Graphing the Equation

Graphing linear equations efficiently requires understanding both intercepts, as they provide key points needed for plotting. Knowing both the x-intercept and y-intercept enables us to sketch the line that represents the equation on a coordinate plane.Here’s how to graph the equation using these intercepts:

• Plot the x-intercept on the graph. For \(3x + 2y = 6\), it's \((2, 0)\).
• Mark the y-intercept on the graph. In our example, it's \((0, 3)\).
• Draw a straight line through these two points. The line extends across the plane, showing all solutions of the equation.

By connecting the intercepts, students can easily visualize how the line behaves and interacts with the axes. Graphing becomes less overwhelming when approached by marking intercepts clearly first. This method highlights the simplicity and beauty of linear equations.