- Set the two expressions for \(y\) equal to solve for \(x\): \(-\frac{1}{3}x + 2 = x - 2\).
- Simplify and solve to find the intersection x-coordinate: \(x = 3\).
- Calculate the corresponding y-coordinate using one of the equations: \(y = 3 - 2 = 1\).
- The solution to the system is the intersection point \(\boxed{(3, 1)}\).
Explanation
- Problem Analysis
To find the intersection point of the two given lines, we analyze the system of equations:
1) \(y = -\frac{1}{3}x + 2\)
2) \(y = x - 2\)
A solution to a system of two linear equations corresponds to the coordinates \((x, y)\) where the two lines intersect. This point must satisfy both equations simultaneously.
- Solving the System of Equations
To solve for the intersection point, we set the two expressions for \(y\) equal to each other because at the intersection point, the y-values are identical:
\[-\frac{1}{3}x + 2 = x - 2\]
To isolate \(x\), we first add 2 to both sides:
\[-\frac{1}{3}x + 4 = x\]
Next, we add \(\frac{1}{3}x\) to both sides:
\[4 = x + \frac{1}{3}x\]
Combining the \(x\) terms, we get:
\[4 = \frac{4}{3}x\]
Multiplying both sides by \(\frac{3}{4}\), we find:
\[x = 4 \times \frac{3}{4} = 3\]
Now, we substitute \(x = 3\) into the second equation to find \(y\):
\[y = 3 - 2 = 1\]
Thus, the point of intersection is \((3, 1)\).
- Identifying the Correct Graph
To identify the correct graph, we look for two specific lines:
- The first line, \(y = -\frac{1}{3}x + 2\), must have a y-intercept at \((0, 2)\) and a downward slope of \(-\frac{1}{3}\) (moving 3 units right for every 1 unit down).
- The second line, \(y = x - 2\), must have a y-intercept at \((0, -2)\) and an upward slope of \(1\) (moving 1 unit right for every 1 unit up).
The correct graph will show these two lines intersecting at the point \((3, 1)\).
Examples
Understanding the intersection of two linear equations is fundamental in fields like economics and logistics. For example, if you are comparing two delivery services, one might have a higher starting fee but a lower cost per mile, while another has a lower starting fee but a higher cost per mile. The point where the total costs of both services are equal, called the break-even point, is found by solving the system of their cost equations. This helps businesses and individuals make data-driven decisions on which service to choose based on the total distance or quantity of items.