Solving Systems of Linear Equations with Fractions

When faced with a system of linear equations that contain fractions, solving them can seem daunting. However, by using substitution and algebraic manipulation, we can simplify the process. This article will guide you through the steps to solve a specific system of equations and highlight the importance of substitution and simplification techniques.

System of Equations with Fractions

Consider the following system of equations:

[frac{11}{2x - 3y} frac{18}{3x - 2y} 13]

[frac{27}{3x - 2y} - frac{2}{2x - 3y} 1]

Step-by-Step Solution

Step 1: Introduce Substitutions for Simplification

To simplify the equations, we can introduce substitutions:

Let (A 2x - 3y) and (B 3x - 2y).

Now, we rewrite the equations in terms of (A) and (B):

[frac{11}{A} frac{18}{B} 13]

[frac{27}{B} - frac{2}{A} 1]

Step 2: Isolate (A) and (B)

From the first equation, solve for (frac{1}{A}):

[frac{11}{A} 13 - frac{18}{B}]

Multiplying through by (AB) (assuming (A eq 0) and (B eq 0), yields:

[11B 13B - 18A]

Rearranging, we get:

[2B 18A]

[B 9A]

Step 3: Substitute (B 9A) into the Second Equation

Substitute (B 9A) into the second equation:

[frac{27}{9A} - frac{2}{A} 1]

This simplifies to:

[frac{3}{A} - frac{2}{A} 1]

[frac{1}{A} 1]

[A 1]

Step 4: Find (B)

Now substitute (A 1) back into (B 9A):

[B 9 times 1 9]

Step 5: Substitute Back to Find (x) and (y)

Now we have:

[A 2x - 3y 1]

[B 3x - 2y 9]

Express (x) from the first equation:

[2x - 3y 1]

[2x 3y 1]

[x frac{3y 1}{2}]

Substitute (x) into the second equation:

[3left(frac{3y 1}{2}right) - 2y 9]

Multiplying through by 2 to eliminate the fraction:

[3(3y 1) - 4y 18]

Simplifying further:

[9y 3 - 4y 18]

[5y 15]

[y 3]

Substitute (y 3) back into the expression for (x):

[x frac{3(3) 1}{2} frac{9 1}{2} frac{10}{2} 5]

Final Solution

The solution to the system of equations is:

[x 5, y 3]

Additional Notes

For systems of equations involving fractions, substitution is a powerful technique to simplify the process. By breaking down the problem into smaller, manageable parts, you can more easily solve for the variables. Another approach, which involves cross-multiplication and elimination, can also be effective but may become more complex with multiple fractions and higher-order polynomials.

Understanding these techniques is crucial for solving a wide range of problems in algebra, particularly when dealing with systems of linear equations with fractions.