Solving Systems of 3 Equations by Substitution

In algebra, a system of equations is a set of two or more equations with the same variables. Solving a system of equations means finding the values of the variables that satisfy all the equations in the system. One common method for solving systems of equations is the substitution method.

Understanding the Substitution Method

The substitution method involves the following steps:

1. Solve one equation for one variable in terms of the other variables. This means isolating the variable on one side of the equation.
2. Substitute the expression you found in step 1 into the other equation(s). This will eliminate one of the variables, resulting in an equation with fewer variables.
3. Solve the new equation for the remaining variable.
4. Substitute the value you found in step 3 back into any of the original equations to solve for the other variable.
5. Check your solution by substituting the values you found into all the original equations.

Example: Solving a System of 3 Equations

Let's consider the following system of three equations:

Equation 1: x + y + z = 6

Equation 2: 2x - y + 3z = 10

Equation 3: 3x + 2y - z = 5

Here's how to solve this system using the substitution method:

1. Solve Equation 1 for x:
   x = 6 - y - z
2. Substitute this expression for x into Equations 2 and 3:
   Equation 2: 2(6 - y - z) - y + 3z = 10
   Equation 3: 3(6 - y - z) + 2y - z = 5
3. Simplify Equations 2 and 3:
   Equation 2: 12 - 2y - 2z - y + 3z = 10
   Equation 3: 18 - 3y - 3z + 2y - z = 5
4. Solve Equation 2 for y:
   -3y + z = -2
   y = (z + 2) / 3
5. Substitute this expression for y into Equation 3:
   18 - 3((z + 2) / 3) - 3z + 2((z + 2) / 3) - z = 5
6. Simplify Equation 3 and solve for z:
   18 - z - 2 - 3z + (2z + 4) / 3 - z = 5
   18 - 5z + (2z + 4) / 3 = 5
   54 - 15z + 2z + 4 = 15
   -13z = -43
   z = 43 / 13
7. Substitute the value of z back into the expression for y:
   y = (43 / 13 + 2) / 3
   y = 49 / 39
8. Substitute the values of y and z back into Equation 1 to solve for x:
   x + 49 / 39 + 43 / 13 = 6
   x = 6 - 49 / 39 - 43 / 13
   x = 107 / 39
9. Check the solution by substituting the values of x, y, and z into all three original equations.
   Equation 1: (107 / 39) + (49 / 39) + (43 / 13) = 6
   Equation 2: 2(107 / 39) - (49 / 39) + 3(43 / 13) = 10
   Equation 3: 3(107 / 39) + 2(49 / 39) - (43 / 13) = 5

Therefore, the solution to the system of equations is x = 107 / 39, y = 49 / 39, and z = 43 / 13.

Advantages of the Substitution Method

• Systematic Approach: The substitution method provides a clear and organized way to solve systems of equations.
• Flexibility: It can be used to solve systems with any number of variables.
• Relatively Easy: The method is generally straightforward, especially for systems with simple equations.

Important Considerations

• Choose the Easiest Variable to Isolate: Select the variable that is easiest to isolate in terms of the other variables. This can simplify the process.
• Be Careful with Signs: Pay attention to the signs of the variables when substituting and solving. Mistakes with signs can lead to incorrect solutions.
• Check Your Solution: Always check your solution by substituting the values back into the original equations to ensure they satisfy all equations in the system.

By following these steps and being mindful of the considerations, you can effectively solve systems of 3 equations by substitution.