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

Jasonbradley

4 min read

·

Jun 16, 2024

35

Share

Solving a System of Linear Equations with Two Variables

This article will discuss how to solve a system of linear equations with two variables, represented by the following equations:

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

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

We can solve this system using various methods, including substitution, elimination, and matrix methods. Here, we will focus on the elimination method.

Elimination Method

The elimination method aims to eliminate one of the variables by manipulating the equations to create opposite coefficients for that variable. Then, by adding the equations together, we can solve for the remaining variable.

Step 1: Identify a variable to eliminate

In this system, we can easily eliminate either 'x' or 'y' by multiplying the equations by appropriate constants. Let's choose to eliminate 'x'.

Step 2: Multiply equations to get opposite coefficients for 'x'

• Multiply the first equation by (a - 2b): (a - 2b)(a + 2b)x + (a - 2b)(2a - b)y = 2(a - 2b)

• Multiply the second equation by -(a + 2b): -(a + 2b)(a - 2b)x - (a + 2b)(2a + b)y = -3(a + 2b)

Step 3: Add the modified equations together

Notice that the 'x' terms now have opposite coefficients and will cancel out when added. We are left with:

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

Step 4: Simplify and solve for 'y'

Expanding and simplifying the equation:

(4a^2 - 5ab + 2b^2)y - (4a^2 + 3ab + 2b^2)y = -a - 8b -8aby = -a - 8b y = (a + 8b) / 8ab

Step 5: Substitute the value of 'y' into one of the original equations and solve for 'x'

Let's substitute 'y' into the first original equation:

(a + 2b)x + (2a - b)(a + 8b) / 8ab = 2

Simplifying and solving for 'x':

(a + 2b)x + (2a^2 + 15ab - 8b^2) / 8ab = 2 (a + 2b)x = 2 - (2a^2 + 15ab - 8b^2) / 8ab (a + 2b)x = (16ab - 2a^2 - 15ab + 8b^2) / 8ab (a + 2b)x = (ab + 8b^2 - 2a^2) / 8ab x = (ab + 8b^2 - 2a^2) / 8ab(a + 2b)

Solution

Therefore, the solution to the system of equations is:

• x = (ab + 8b^2 - 2a^2) / 8ab(a + 2b)
• y = (a + 8b) / 8ab

This solution can be expressed in terms of 'a' and 'b', and it represents the intersection point of the two linear equations when graphed.