- Find the x-intercept by setting \(y=0\), which results in \((4, 0)\).
- Find the y-intercept by setting \(x=0\), which results in \((0, -3)\).
- Plot these two points on the Cartesian coordinate system.
- Connect the points with a straight line to visualize the equation \(6x - 8y = 24\).
- Final intercepts are \(\boxed{(4,0)}\) and \(\boxed{(0,-3)}\).
Explanation
- Problem Analysis
To graph the linear equation \(6x - 8y = 24\), we begin by identifying its intercepts, which are the points where the line crosses the coordinate axes. The x-intercept occurs where the line crosses the x-axis, at which point the y-coordinate is \(0\). The y-intercept occurs where the line crosses the y-axis, at which point the x-coordinate is \(0\). Determining these points provides a simple and efficient way to draw the line.
- Finding the Intercepts
To find the x-intercept, we set \(y = 0\) in the equation \(6x - 8y = 24\):
\[6x - 8(0) = 24\]
\[6x = 24\]
\[x = 4\]
Thus, the x-intercept is the point \((4, 0)\).
To find the y-intercept, we set \(x = 0\) in the equation \(6x - 8y = 24\):
\[6(0) - 8y = 24\]
\[-8y = 24\]
\[y = -3\]
Thus, the y-intercept is the point \((0, -3)\).
- Graphing the Line
With the points \((4, 0)\) and \((0, -3)\) identified, we can now graph the line. We place a point at \(x=4\) on the horizontal x-axis and a point at \(y=-3\) on the vertical y-axis. By drawing a straight line through these two points, we obtain the graphical representation of the equation \(6x - 8y = 24\). This line extends infinitely in both directions, representing all pairs of \((x, y)\) that satisfy the given equation.
Examples
Linear equations are fundamental in fields like economics and engineering for modeling relationships between two variables. For example, if you are planning a budget, you might have a fixed cost and a variable cost related to two different services. The equation \(6x - 8y = 24\) could represent the boundary of a budget where \(x\) and \(y\) are the amounts of two different resources you can afford. By finding the intercepts, you quickly determine the maximum amount of one resource you can purchase if you choose to spend nothing on the other, helping you visualize the trade-offs available within your constraints.