- Set the two equations equal to solve for \(x\): \(-2x + 4 = -\frac{1}{3}x - 1\).
- Simplify the equation to find \(x = 3\).
- Substitute \(x = 3\) into either equation to determine \(y = -2\).
- State the solution as the ordered pair \(\boxed{(3, -2)}\).
Explanation
- Understanding the Problem
The problem asks us to find the solution to the system of linear equations:
\[ \begin{cases} y = -2x + 4 \ y = -\frac{1}{3}x - 1 \end{cases} \]
The solution to a system of equations is the ordered pair \((x, y)\) that satisfies both equations simultaneously. Geometrically, this represents the point where the two lines intersect on a Cartesian coordinate system. Given both equations are in slope-intercept form (\(y = mx + b\)), we can find the point of intersection by setting the expressions for \(y\) equal to each other.
- Solving the System of Equations
To find the intersection point, we set the two expressions equal:
\[-2x + 4 = -\frac{1}{3}x - 1\]
Next, we isolate the variable \(x\). Adding \(2x\) to both sides and adding \(1\) to both sides gives:
\[4 + 1 = 2x - \frac{1}{3}x\]
\[5 = \left(\frac{6}{3} - \frac{1}{3}\right)x\]
\[5 = \frac{5}{3}x\]
Multiplying both sides by \(\frac{3}{5}\), we find:
\[x = 5 \times \frac{3}{5} = 3\]
Now, substitute \(x = 3\) back into the first equation to find \(y\):
\[y = -2(3) + 4 = -6 + 4 = -2\]
Thus, the intersection point is \((3, -2)\).
- Conclusion and Verification
We can verify the intersection point by substituting \(x = 3\) into the second equation:
\[y = -\frac{1}{3}(3) - 1 = -1 - 1 = -2\]
Since both equations result in \(y = -2\) when \(x = 3\), the point \((3, -2)\) is indeed the solution to the system. The lines intersect at exactly one point, confirming that this is the unique solution to the system. The final solution is \(\boxed{(3, -2)}\).
Examples
Systems of linear equations are used in many real-world scenarios, such as comparing the costs of two different service providers. For example, if Provider A charges a fixed fee of \(\$4\) with a \(\$2\) rate per unit, while Provider B charges a \(\$1\) fee but with a \(\$1/3\) rate per unit, you can model these costs as \(y = 2x + 4\) and \(y = 1/3x + 1\). Finding where these lines intersect helps determine the break-even point—the exact number of units where the cost for both services becomes equal. Understanding these systems helps in making informed financial and resource-allocation decisions.