- Set the two equations \(-x = 2x + 6\) equal to each other to solve for \(x\).
- Calculate \(x = -2\) by algebraic manipulation.
- Substitute \(x = -2\) into the original equations to find \(y = 2\).
- Identify the intersection point as \((-2, 2)\), which is option a.
- \(\boxed{a}\)
Explanation
- Analyzing the System of Equations
To find the point of intersection of the two lines, we need to solve the system of equations where both equations are equal. The system is given by:
\[ \begin{cases} y = -x \ y = 2x + 6 \end{cases} \]
An intersection point \((x, y)\) is a point that satisfies both equations simultaneously. Since both expressions are equal to \(y\), we can set the right-hand sides of the equations equal to each other.
- Solving for x
We set the two expressions for \(y\) equal:
\[-x = 2x + 6\]
To solve for \(x\), we first subtract \(2x\) from both sides of the equation:
\[-x - 2x = 6\]
\[-3x = 6\]
Now, divide both sides by \(-3\):
\[x = \frac{6}{-3} = -2\]
So, the x-coordinate of the intersection point is \(-2\).
- Solving for y and Verifying the Solution
Now that we have the x-coordinate, we substitute \(x = -2\) into the first equation, \(y = -x\), to find the corresponding y-coordinate:
\[y = -(-2) = 2\]
Alternatively, substituting \(x = -2\) into the second equation, \(y = 2x + 6\), gives:
\[y = 2(-2) + 6 = -4 + 6 = 2\]
Since both equations yield the same y-coordinate, the point of intersection is \((-2, 2)\). This matches option a.
Examples
Imagine you are tracking two objects moving in a straight path across a coordinate grid. One object follows the path \(y = -x\), and the other follows the path \(y = 2x + 6\). Finding the intersection point allows you to determine exactly where and when these two objects will collide or cross each other's path. This is a fundamental concept in fields like navigation, robotics, and physics, where understanding the relationship between linear paths is crucial for safety and coordination.