(2(x+4))/3-y/2=9/2x+2y-1/3*(3x-2)=-4/3

3 min read Jun 16, 2024

Solving a System of Linear Equations

This article will guide you through the process of solving a system of linear equations. We will be working with the following system:

Equation 1: (2(x+4))/3 - y/2 = 9/2 Equation 2: 2y - 1/3*(3x-2) = -4/3

Step 1: Simplify the Equations

Equation 1:
- Distribute the 2: (2x + 8)/3 - y/2 = 9/2
- Find a common denominator for the fractions: (4x + 16)/6 - 3y/6 = 27/6
- Combine the terms on the left side: (4x - 3y + 16)/6 = 27/6
- Multiply both sides by 6: 4x - 3y + 16 = 27
- Simplify: 4x - 3y = 11
Equation 2:
- Distribute the -1/3: 2y - x + 2/3 = -4/3
- Find a common denominator for the fractions: 6y/3 - 3x/3 + 2/3 = -4/3
- Combine the terms on the left side: (-3x + 6y + 2)/3 = -4/3
- Multiply both sides by 3: -3x + 6y + 2 = -4
- Simplify: -3x + 6y = -6

Step 2: Choose a Method to Solve

We have two main methods for solving a system of linear equations:

Substitution: Solve one equation for one variable and substitute it into the other equation.
Elimination: Multiply one or both equations by constants to make the coefficients of one variable opposites. Then add the equations together to eliminate that variable.

Let's use the elimination method in this case.

Step 3: Eliminate One Variable

Notice that the coefficients of x in the two equations are already opposites (4x and -3x).
Add the two simplified equations together: (4x - 3y = 11) + (-3x + 6y = -6)
This eliminates x, leaving: 3y = 5

Step 4: Solve for the Remaining Variable

Step 5: Substitute to Find the Other Variable

Substitute y = 5/3 into either of the original simplified equations. Let's use 4x - 3y = 11: 4x - 3(5/3) = 11 4x - 5 = 11
Add 5 to both sides: 4x = 16
Divide both sides by 4: x = 4

Solution:

The solution to the system of equations is x = 4 and y = 5/3.