How to Solve a System by the Substitution Method

Oct 13, 2025 by magicalceylontours.com 81 views

**Solve the System by the Substitution Method** =====================================================

What is the Substitution Method?

The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other. This method is useful when one of the equations is already solved for one of the variables.

How to Solve a System by the Substitution Method

To solve a system of linear equations by the substitution method, follow these steps:

Identify the equations: Write down the two linear equations that make up the system.
Solve one equation for one variable: Choose one of the equations and solve it for one of the variables. This will give you an expression for that variable in terms of the other variable.
Substitute the expression into the other equation: Take the expression you obtained in step 2 and substitute it into the other equation.
Solve for the remaining variable: Simplify the resulting equation and solve for the remaining variable.
Back-substitute to find the other variable: Once you have found the value of one variable, substitute it back into one of the original equations to find the value of the other variable.

Example: Solving a System by the Substitution Method

Let's consider the following system of linear equations:

{ \begin{aligned} x + 2y &= 6 \\ y &= 2x + 3 \end{aligned} \}

To solve this system by the substitution method, we will follow the steps outlined above.

Step 1: Identify the equations

The two linear equations that make up the system are:

$x + 2y = 6$

$y = 2x + 3$

Step 2: Solve one equation for one variable

We will solve the second equation for the variable $y$ :

$y = 2x + 3$

Step 3: Substitute the expression into the other equation

We will substitute the expression for $y$ into the first equation:

$x + 2(2x + 3) = 6$

Step 4: Solve for the remaining variable

Simplifying the resulting equation, we get:

$x + 4x + 6 = 6$

Combine like terms:

$5x + 6 = 6$

Subtract 6 from both sides:

$5x = 0$

Divide both sides by 5:

$x = 0$

Step 5: Back-substitute to find the other variable

Now that we have found the value of $x$ , we can substitute it back into one of the original equations to find the value of $y$ :

$y = 2x + 3$

$y = 2(0) + 3$

$y = 3$

Therefore, the solution to the system is $(x, y) = (0, 3)$ .

Q&A

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other.

Q: How do I know which equation to solve for one variable?

A: You can choose either equation to solve for one variable. The key is to make sure that you have an expression for one variable in terms of the other variable.

Q: What if I get stuck during the substitution process?

A: If you get stuck during the substitution process, try simplifying the equation or using a different method to solve the system.

Q: Can I use the substitution method to solve a system with more than two equations?

A: No, the substitution method is typically used to solve a system of two linear equations. If you have a system with more than two equations, you may need to use a different method, such as the elimination method or the matrix method.

Q: What is the advantage of using the substitution method?

A: The advantage of using the substitution method is that it can be easier to visualize and understand the relationships between the variables in the system.

Q: What is the disadvantage of using the substitution method?

A: The disadvantage of using the substitution method is that it can be more time-consuming and labor-intensive than other methods, such as the elimination method.

Conclusion

The substitution method is a useful technique for solving systems of linear equations. By following the steps outlined above, you can use the substitution method to solve a system of two linear equations. Remember to identify the equations, solve one equation for one variable, substitute the expression into the other equation, solve for the remaining variable, and back-substitute to find the other variable. With practice and patience, you can become proficient in using the substitution method to solve systems of linear equations.