Substitution Method for Solving Linear Equations in Two Variables

The substitution method is a technique used to solve linear equations in two variables. It involves substituting the value of one variable in terms of the other variable in one equation and solving for the remaining variable. This value is then substituted into the other equation to find the value of the second variable. The steps involved in the substitution method are as follows:

Step 1: Solve one equation for one variable
Choose one of the two given equations and solve it for one of the variables in terms of the other variable. This will give you an expression for one of the variables.

Step 2: Substitute the expression into the other equation
Substitute the expression obtained from step 1 into the other equation. This will give you an equation with only one variable.

Step 3: Solve for the variable
Solve the equation obtained from step 2 for the variable.

Step 4: Substitute the value back into the expression
Substitute the value obtained from step 3 into the expression obtained from step 1 to find the value of the other variable.

Example of the Substitution Method

Let us consider the following system of equations:
2x + 3y = 9
4x - 5y = - 10

Step 1: Solve one equation for one variable
We can solve the first equation for x as follows:
2x + 3y = 9
2x = 9 - 3y
x = (9 - 3y)/2

Step 2: Substitute the expression into the other equation
Substitute the expression for x obtained in step 1 into the second equation:
4x - 5y = -10
4((9 - 3y)/2) - 5y = -10

Step 3: Solve for the variable
Simplify and solve for y:
18 - 6y - 5y = -10
-11y = -28
y = 28/11

Step 4: Substitute the value back into the expression
Substitute the value of y obtained in step 3 into the expression for x obtained in step 1:
x = (9 - 3y)/2
x = (9 - 3(28/11))/2
x = 5/11

Therefore, the solution of the system of equations is (5/11, 28/11).

Conclusion

The substitution method is a useful technique for solving linear equations in two variables. It involves substituting the value of one variable in terms of the other variable in one equation and solving for the remaining variable. This value is then substituted into the other equation to find the value of the second variable.

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