Solving Equations for Variables x and y: A Step-by-Step Guide

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