(a+2b)x+(2a-b)y=2 (a-2b)x+(2a+b)y=3 By Cross Multiplication

Jasonbradley

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Jun 16, 2024

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Solving Simultaneous Equations using Cross Multiplication

This article will demonstrate how to solve the following system of simultaneous equations using the cross-multiplication method:

(a + 2b)x + (2a - b)y = 2

(a - 2b)x + (2a + b)y = 3

Understanding the Cross-Multiplication Method

The cross-multiplication method is a technique used to solve a system of two linear equations in two variables. It involves manipulating the coefficients of the variables to eliminate one variable and solve for the other. The steps are as follows:

1. Arrange the equations: Ensure both equations are in the standard form (ax + by = c).
2. Cross-multiply: Multiply the coefficient of x in the first equation by the coefficient of y in the second equation, and vice versa.
3. Subtract the products: Subtract the product obtained in step 2 from the product obtained in step 1.
4. Solve for x: Divide the result from step 3 by the difference of the products of the coefficients of x and y in each equation.
5. Substitute the value of x: Substitute the value of x obtained in step 4 into any of the original equations and solve for y.

Applying the Method to Our Equations

Let's apply the cross-multiplication method to the given system of equations:

1. The equations are already in standard form.
2. Cross-multiply:
   • (a + 2b)(2a + b) - (2a - b)(a - 2b)
3. Subtract the products:
   • 2a² + 5ab + 2b² - (2a² - 5ab + 2b²) = 10ab
4. Solve for x:
   • x = (10ab) / [(a + 2b)(2a + b) - (2a - b)(a - 2b)]
   • x = (10ab) / (10ab) = 1
5. Substitute the value of x: Let's substitute x = 1 into the first equation:
   • (a + 2b)(1) + (2a - b)y = 2
   • a + 2b + (2a - b)y = 2
   • (2a - b)y = 2 - a - 2b
   • y = (2 - a - 2b) / (2a - b)

Therefore, the solution to the system of equations is x = 1 and y = (2 - a - 2b) / (2a - b).

Important Notes

• This method is valid only when the determinant of the coefficient matrix is non-zero. In this case, the determinant is 10ab, which is non-zero unless a or b is zero.
• If the determinant is zero, the system may have no solutions or infinitely many solutions.

This article provides a step-by-step guide to solving a system of simultaneous equations using the cross-multiplication method. By following these steps, you can efficiently solve for the values of the variables. Remember to pay close attention to the signs and coefficients involved in the calculations.