Solving the System of Linear Equations
In this article, we will discuss how to solve a system of linear equations involving fractions. The equations we will be working with are:
Equation 1: 3x - y + 7/11 + 2 = 10 Equation 2: 2y + x + 11/7 = 10
Step 1: Simplify the Equations
Before we start solving the system, let's simplify the equations by multiplying both sides of each equation by the least common multiple (LCM) of the denominators.
Equation 1: 3x - y + 7/11 + 2 = 10 Multiply both sides by 11: 33x - 11y + 7 + 22 = 110
Simplify: 33x - 11y + 29 = 110
Equation 2: 2y + x + 11/7 = 10 Multiply both sides by 7: 14y + 7x + 11 = 70
Simplify: 14y + 7x = 59
Step 2: Solve the System Using Substitution or Elimination
Now we have two linear equations with two variables. We can solve this system using either the substitution method or the elimination method.
Let's use the elimination method. We can eliminate the y-variable by multiplying Equation 1 by 14 and Equation 2 by 11, and then subtracting one equation from the other.
Equation 1 (multiplied by 14): 462x - 154y + 406 = 1540 Equation 2 (multiplied by 11): 154y + 77x = 649
Subtract Equation 2 from Equation 1: 385x = 891
Divide by 385: x = 891/385 x = 231/77
Now that we have the value of x, we can substitute it into one of the original equations to find the value of y.
Substitute x into Equation 2: 14y + 7(231/77) = 59
Simplify: 14y + 161/11 = 59
Multiply both sides by 11: 154y + 161 = 649
Subtract 161 from both sides: 154y = 488
Divide by 154: y = 488/154 y = 244/77
Solution: x = 231/77 y = 244/77
Conclusion
In this article, we successfully solved a system of linear equations involving fractions using the elimination method. We simplified the equations, eliminated one variable, and solved for the other. Finally, we found the values of x and y, which are x = 231/77 and y = 244/77.
