Problem 64 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. $$ 4 x-3 y=6 $$

Step-by-Step Solution

Answer

x-intercept: \( (\frac{3}{2}, 0) \), y-intercept: \( (0, -2) \).

Step 1: Find the x-intercept

To find the x-intercept, we need to set \( y = 0 \) in the equation and solve for \( x \). The equation becomes:\[ 4x - 3(0) = 6 \]\[ 4x = 6 \]\[ x = \frac{6}{4} = \frac{3}{2} \]So, the x-intercept is \( \left( \frac{3}{2}, 0 \right) \).

Step 2: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) in the equation and solve for \( y \). The equation becomes:\[ 4(0) - 3y = 6 \]\[ -3y = 6 \]\[ y = \frac{6}{-3} = -2 \]So, the y-intercept is \( (0, -2) \).

Step 3: Graph the Equation

With the found intercepts, we can plot the x-intercept at \( \left( \frac{3}{2}, 0 \right) \) and the y-intercept at \( (0, -2) \) on a coordinate plane. Draw a straight line through these points to graph the equation \( 4x - 3y = 6 \).

Key Concepts

Understanding X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. This occurs when the y-value is zero, as the x-intercept represents the horizontal position on the graph. To find the x-intercept from a linear equation, simply set the y variable to zero and solve for the x variable.
In the example equation, \(4x - 3y = 6\), we set \(y = 0\) and solve for \(x\):

• Substitute: \(4x - 3(0) = 6\)
• Simplify: \(4x = 6\)
• Solve: \(x = \frac{6}{4} = \frac{3}{2}\)

Hence, the x-intercept for this equation is \(\left( \frac{3}{2}, 0 \right)\). This tells us exactly where the line will touch the x-axis on a graph.

Understanding Y-Intercept

The y-intercept of a graph is the spot where the graph crosses the y-axis. This happens when the x-value is zero, as the y-intercept indicates the vertical position on the graph. To locate the y-intercept from a linear equation, set the x variable to zero and solve for the y variable.
Using the same equation, \(4x - 3y = 6\), set \(x = 0\) and solve for \(y\):

• Substitute: \(4(0) - 3y = 6\)
• Simplify: \(-3y = 6\)
• Solve: \(y = \frac{6}{-3} = -2\)

Thus, the y-intercept is \((0, -2)\). This indicates where the line intersects the y-axis on a graph.

Graphing Equations

Graphing a linear equation involves plotting points and drawing a straight line through them. It's much like connecting dots on a paper. After finding both the x-intercept and y-intercept, these intercepts serve as your guiding points. They define the trajectory of the line on the graph.
With the equation \(4x - 3y = 6\), plot the points you found:

• X-intercept point: \(\left( \frac{3}{2}, 0 \right)\)
• Y-intercept point: \((0, -2)\)

Once these points are plotted on a coordinate plane, simply draw a straight line connecting them. This line represents the graph of the equation. When drawing, ensure the line continues in both directions, from one side of the graph to the other, which signifies that the graph extends infinitely in both directions. In real-world applications, each point on this line represents a solution to the equation, visually showing the relationship between \(x\) and \(y\).