Solving the System of Linear Equations
In this article, we will solve the system of linear equations:
Equation 1: 5/x - 3/y = 1 Equation 2: 3/2x + 2/3y = 5
To solve this system, we will use the method of substitution or elimination. Here, we will use the substitution method.
Step 1: Solve one of the equations for one variable
Let's solve Equation 1 for x:
5/x - 3/y = 1 5/x = 1 + 3/y x = 5 / (1 + 3/y)
Now, we have expressed x in terms of y.
Step 2: Substitute the expression into the other equation
Substitute the expression for x into Equation 2:
3/2(5 / (1 + 3/y)) + 2/3y = 5
Step 3: Simplify and solve for y
Simplify the equation:
15/(2(1 + 3/y)) + 2/3y = 5
Multiply both sides by 6 to eliminate the fractions:
45/(1 + 3/y) + 4y = 30
Now, simplify the equation further:
45 = 30(1 + 3/y) - 4y
45 = 30 + 90/y - 4y
Rearrange the equation:
90/y - 4y = 15
Step 4: Solve for y
y = 3 or y = -15
Step 5: Substitute the values of y back into one of the original equations to find x
Substitute y = 3 into Equation 1:
5/x - 3/3 = 1 5/x - 1 = 1 5/x = 2 x = 5/2
Substitute y = -15 into Equation 1:
5/x - 3/-15 = 1 5/x + 1/5 = 1 5/x = 4/5 x = 25/4
Step 6: Write the final solutions
The solutions to the system of linear equations are:
x = 5/2, y = 3 x = 25/4, y = -15
Therefore, we have found the two solutions to the system of linear equations.
