Solving Systems of Linear Equations with Substitution

In algebra, a system of linear equations is a set of two or more linear equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. One method for solving such systems is the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation.

Steps for Solving Systems of Linear Equations by Substitution

1. 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.
2. Substitute the expression into the other equation. Substitute the expression you just found into the other equation. This will eliminate one of the variables, leaving you with an equation in only one variable.
3. Solve the resulting equation. Solve the equation you obtained in step 2 for the remaining variable. This will give you the value of one of the variables.
4. Substitute the value back into one of the original equations. Substitute the value you just found back into one of the original equations (either of the two you started with). This will allow you to solve for the other variable.
5. Check your solution. Substitute the values you found for both variables back into both original equations to ensure that they satisfy both equations simultaneously.

Example

Let's solve the following system of equations using the substitution method:

```
2x + y = 5

x - 3y = 1
```

1. **Solve for one variable:** We can solve the second equation for *x*:
```
x = 3y + 1
```

2. **Substitute:** Now, substitute this expression for *x* into the first equation:
```
2(3y + 1) + y = 5
```

3. **Solve the resulting equation:** Simplify and solve for *y*:
```
6y + 2 + y = 5
7y = 3
y = 3/7
```

4. **Substitute back:** Substitute the value of *y* back into the equation we solved for *x*:
```
x = 3(3/7) + 1
x = 9/7 + 1
x = 16/7
```

5. **Check:** Now, let's check our solution by substituting both values back into the original equations:
```
2(16/7) + 3/7 = 5
32/7 + 3/7 = 5
35/7 = 5
5 = 5 (True)

16/7 - 3(3/7) = 1
16/7 - 9/7 = 1
7/7 = 1
1 = 1 (True)
```

Therefore, the solution to the system of equations is *x = 16/7* and *y = 3/7*.

Applications of Substitution Method

The substitution method is a widely used technique in various fields, including:

* **Mathematics:** Solving systems of equations in algebra, calculus, and linear algebra.
* **Physics:** Modeling physical systems and solving for unknown quantities.
* **Engineering:** Designing and analyzing structures, circuits, and other systems.
* **Economics:** Analyzing economic models and predicting market behavior.

By understanding the substitution method, you gain a powerful tool for solving problems in various disciplines. It's a fundamental concept in algebra and has applications in numerous real-world situations.