Verified Answer :
- Set the two linear equations equal to each other to solve for the x-coordinate: \(-x + 6 = \frac{2}{3}x - 9\).
- Calculate the x-coordinate as \(x = 9\).
- Substitute the value of \(x\) back into one of the original equations to find the y-coordinate: \(y = -(9) + 6 = -3\).
- Conclude that the intersection point of the lines is \(\boxed{(9, -3)}\).
Explanation
- Analysis of the Problem
The objective is to solve the system of linear equations:
1) \(y = -x + 6\)
2) \(y = \frac{2}{3}x - 9\)
These equations represent two straight lines in a Cartesian plane. Finding the solution means finding the coordinate pair \((x, y)\) where these two lines intersect. This point must satisfy both equations simultaneously. We will use algebraic substitution to find this point and then describe how it can be visualized.
- Algebraic Solution
To find the point of intersection, we set the two expressions for \(y\) equal to each other, since at the intersection point, their \(y\)-values must be identical:
\[-x + 6 = \frac{2}{3}x - 9\]
Next, we isolate \(x\) by adding \(x\) to both sides and adding \(9\) to both sides:
\[6 + 9 = \frac{2}{3}x + x\]
\[15 = \frac{5}{3}x\]
To solve for \(x\), we multiply both sides by the reciprocal of \(\frac{5}{3}\), which is \(\frac{3}{5}\):
\[x = 15 \times \frac{3}{5}\]
\[x = 9\]
Now that we have the value for \(x\), we substitute it back into the first equation to find \(y\):
\[y = -(9) + 6\]
\[y = -3\]
Thus, the point of intersection is \((9, -3)\).
- Verification and Visualization
We verify the solution \((9, -3)\) by substituting it into both original equations:
For the first equation: \(-9 + 6 = -3\), which is correct.
For the second equation: \(\frac{2}{3}(9) - 9 = 6 - 9 = -3\), which is also correct.
Graphically, the first line passes through \((0, 6)\) and \((6, 0)\), while the second line passes through \((0, -9)\) and \((3, -7)\). Plotting these lines on a coordinate plane will show them crossing exactly at \((9, -3)\).
- Conclusion
We have determined that the system of equations is consistent and independent, with the unique solution being the intersection point of the two lines at \((9, -3)\). Therefore, the final answer is \(\boxed{(9, -3)}\).
Examples
Systems of equations are fundamental in fields like logistics and economics to find "equilibrium points." For instance, imagine a company selling a product where the cost to produce is modeled by one linear equation, and the total revenue from sales is modeled by another. The point where the two lines intersect represents the "break-even point," where the total revenue equals the total costs. Calculating this point tells the business exactly how many units must be sold to neither lose nor gain money, serving as a critical threshold for decision-making.