Solving Systems of Equations by Substitution: A Comprehensive Guide

In this article, we will explore the substitution method for solving systems of equations. This method involves finding a variable as a function of the other in one equation and then substituting that function into the other equation to solve for the single variable.

Understanding Systems of Equations

A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for these variables that satisfy all the equations simultaneously. Typically, a system of equations can have one unique solution, no solution, or infinitely many solutions depending on the equations.

The Substitution Method

The substitution method is one of the most straightforward techniques for solving systems of equations. Here's a step-by-step guide:

Identify the variables and equations. Choose one equation and solve it for one of the variables. Substitute the expression for the solved variable into the other equation. Solve the resulting equation for the remaining variable. Substitute the value of the solved variable back into the expression you found in step 2. Verify the solution by substituting the values into the original equations.

Step-by-Step Example

Consider the following system of equations:

Equation 1: (2x 3y 5)

Equation 2: (xy 2)

Step 1: From Equation 2, we can express (y) as a function of (x): (y 2 - x).

Substitution

Step 2: Substitute (y 2 - x) into Equation 1:

(2x 3(2 - x) 5)

Step 3: Simplify and solve for (x):

(2x 6 - 3x 5)

(-x 6 5)

(-x -1)

(x 1)

Back Substitution

Step 4: Substitute (x 1) back into (y 2 - x):

(y 2 - 1 1)

Verification

Step 5: Verify the solution by substituting (x 1) and (y 1) into the original equations:

For Equation 1: (2(1) 3(1) 2 3 5) For Equation 2: (1(1) 1 2)

Additional Example

Consider another system of equations for further clarity:

Equation 1: (xy 105)

Equation 2: (xy 22)

Step 2: From Equation 2, we can express (x) as a function of (y): (x 22 - y).

Substitution

Step 3: Substitute (x 22 - y) into Equation 1:

((22 - y)y 105)

(22y - y^2 105)

(y^2 - 22y 105 0)

Step 4: Solve the quadratic equation:

(y^2 - 17y - 5y 105 0)

(y(y - 17) - 5(y - 17) 0)

((y - 17)(y - 5) 0)

(y 17) or (y 5)

Back Substitution

Step 5: Substitute (y 17) and (y 5) back into (x 22 - y):

(x 22 - 17 5) (when (y 17))

(x 22 - 5 17) (when (y 5))

Final Solutions

The solutions to the system of equations are:

((x, y) (5, 17)) ((x, y) (17, 5))

General Method

If the equations are of the form (ax by c) and (a'x b'y d), solve any one of them for one variable. For example, solve for (x):

(x frac{c - by}{a})

Then substitute the value of (x) found in the second equation to solve for (y).

Conclusion

Mastery of the substitution method can be highly beneficial in solving complex systems of equations. By following these steps and practicing with various examples, you can enhance your problem-solving skills in algebra.