Solving Systems of Linear Equations using Substitution Method
In this article, we will solve the following system of linear equations using the substitution method:
$\frac{2x}{a} + \frac{y}{b} = 2 ... (1)$ $\frac{x}{a} - \frac{y}{b} = 4 ... (2)$
Step 1: Solve one of the equations for one variable
We will solve equation (2) for $y$ in terms of $x$. Multiplying both sides of equation (2) by $b$, we get:
$\frac{bx}{a} - y = 4b$
Now, solving for $y$, we get:
$y = \frac{bx}{a} - 4b ... (3)$
Step 2: Substitute the expression into the other equation
Substituting the expression for $y$ from equation (3) into equation (1), we get:
$\frac{2x}{a} + \frac{\frac{bx}{a} - 4b}{b} = 2$
Simplifying the equation, we get:
$\frac{2x}{a} + \frac{bx}{ab} - \frac{4b}{b} = 2$
$\frac{2x}{a} + \frac{x}{a} - 4 = 2$
Step 3: Solve for the variable
Combining like terms, we get:
$\frac{3x}{a} - 4 = 2$
Adding 4 to both sides, we get:
$\frac{3x}{a} = 6$
Multiplying both sides by $a$, we get:
$3x = 6a$
Dividing both sides by 3, we get:
$x = 2a$
Step 4: Substitute the value back into one of the original equations to find the other variable
Substituting the value of $x$ into equation (3), we get:
$y = \frac{b(2a)}{a} - 4b$
Simplifying, we get:
$y = 2b - 4b$
$y = -2b$
Therefore, the solution to the system of linear equations is:
$x = 2a$ $y = -2b$
In conclusion, we have successfully solved the system of linear equations using the substitution method.
