Solving Equations for Variables x and y: A Step-by-Step Guide
In algebra, solving simultaneous equations is a fundamental skill used in various mathematical and real-world applications. This guide will walk you through the process of solving the equations x/31x-1 y-1/3y1 to find the values of x and y.
Understanding the Problem
The given equation is:
(frac{x}{3}1 y - frac{1}{3}y1)
Step 1: Isolate Variables
First, we rewrite the equation:
(frac{x}{3}1 - y frac{1}{3}y1 0)
Next, we can simplify the equation to:
(frac{x-1}{3} frac{1}{3}(y-1) 0)
Step 2: Simplify the Equation
Multiply both sides by 3 to clear the fraction:
(x-1 y-1 0)
This simplifies to:
(x y -2 0)
Therefore, we have:
(x y 2)
Step 3: Solve the Simultaneous Equations
Given the equations:
(frac{x-1}{3} y - frac{1}{3}(y-1)) (x - 1 y 1)We start by solving the second equation for y:
(x - 1 y 1)
(y x - 2)
Substitute (y x - 2) into the first equation:
(frac{x-1}{3} (x-2) - frac{1}{3}(x-3))
Simplify the right-hand side:
(frac{x-1}{3} x - 2 - frac{x-3}{3})
Combine like terms:
(frac{x-1}{3} frac{3(x-2) - (x-3)}{3})
(frac{x-1}{3} frac{3x - 6 - x 3}{3})
(frac{x-1}{3} frac{2x - 3}{3})
Multiply both sides by 3 to eliminate the denominator:
(x - 1 2x - 3)
Rearrange to solve for x:
(x - 2x -3 1)
(-x -2)
(x 2)
Step 4: Find the Value of y
Using the value of (x 5), substitute back into (y x - 2):
(y 5 - 2)
(y 3)
Verification
To verify, substitute (x 5) and (y 3) back into the original equations:
(frac{x}{3} - 1 y - frac{1}{3}y - 1)
(frac{5}{3} - 1 3 - frac{1}{3}3 - 1)
(2 - frac{2}{3} 2 - frac{2}{3})
(2frac{1}{3} 2frac{1}{3})
Both equations are satisfied, thus the values are correct.
Conclusion
The values of x and y are (x 5) and (y 3).
Additional Resources
For more detailed explanations and practice on solving simultaneous equations, consider the following resources:
Textbook on Algebra Interactive Practice Questions Video Tutorials on Solving Equations