- The vertical line \(x = 9\) is graphed as a straight line passing through \(x=9\).
- The linear equation \(3x - 2y = 9\) is rearranged to the slope-intercept form \(y = \frac{3}{2}x - 4.5\).
- By substituting \(x=9\) into the equation \(3x - 2y = 9\), we find the intersection point.
- The system has the unique solution \(\boxed{(9, 9)}\).
Explanation
- Understanding the System of Equations
The problem asks us to solve the system of linear equations:
\[x = 9\]
\[3x - 2y = 9\]
The equation \(x = 9\) defines a vertical line where every point has an x-coordinate of 9. This line is parallel to the y-axis and intersects the x-axis at \((9, 0)\). The second equation, \(3x - 2y = 9\), is a linear equation that can be graphed by finding two points or by converting it into the slope-intercept form, \(y = mx + b\). The objective is to identify the intersection point of these two lines on the Cartesian coordinate plane.
- Graphing the Lines
To graph the second equation, \(3x - 2y = 9\), we rearrange it into slope-intercept form:
\[-2y = -3x + 9\]
Dividing both sides by -2:
\[y = \frac{3}{2}x - 4.5\]
This line has a slope of \(\frac{3}{2}\) (it rises 3 units for every 2 units it runs to the right) and a y-intercept of \(-4.5\). We can pick two points to plot this line:
- If \(x = 1\), \(y = \frac{3}{2}(1) - 4.5 = 1.5 - 4.5 = -3\). Point: \((1, -3)\).
- If \(x = 5\), \(y = \frac{3}{2}(5) - 4.5 = 7.5 - 4.5 = 3\). Point: \((5, 3)\).
By drawing a line through these two points and the vertical line \(x = 9\), we can observe where they meet.
- Finding the Point of Intersection
To find the exact point of intersection algebraically, we substitute the value of \(x\) from the first equation into the second equation:
\[3(9) - 2y = 9\]
\[27 - 2y = 9\]
Subtract 27 from both sides:
\[-2y = 9 - 27\]
\[-2y = -18\]
Divide by -2:
\[y = 9\]
Thus, the intersection point of the two lines is \((9, 9)\). This confirms that both lines pass through the point \((9, 9)\) on the coordinate plane.
Examples
Imagine you are planning the layout of a park. You want to place a central fountain on a straight path that runs from west to east represented by a specific coordinate line, while another walking trail runs across the park described by a linear equation. By graphing these two paths on a coordinate grid, you can find the exact intersection point, which tells you precisely where the fountain should be placed to be accessible from both paths. This method is fundamental in urban planning, logistics, and navigation, where identifying the meeting point of two distinct routes is crucial.