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 of the equations in the system. One common method for solving systems of equations is the substitution method.

What is the Substitution Method?

The substitution method involves solving one equation for one variable in terms of the other variables and then substituting that expression into the other equations. This process eliminates one variable from the system and reduces the number of equations. The goal is to eventually get down to a single equation with one unknown, which can then be solved directly.

Steps for Solving a System of 3 Equations by Substitution

Here's a step-by-step guide to solve a system of three equations using the substitution method:

1. Choose an Equation and Variable: Select one of the equations and choose a variable that appears in that equation. Solve this equation for the chosen variable in terms of the other variables.
2. Substitute: Substitute the expression you just found for the chosen variable into the other two equations. This will give you a new system of two equations with two unknowns.
3. Solve the New System: Solve the new system of two equations using the substitution method again or any other method you prefer (e.g., elimination). You'll now have the values for two of the variables.
4. Back-Substitute: Substitute the values you found for the two variables back into the original equation you used in step 1. This will give you the value for the third variable.
5. Check Your Solution: Substitute the values you found for all three variables into each of the original equations to ensure they are all satisfied. This verifies your solution.

Example:

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

Equation 1: x + 2y - z = 1

Equation 2: 2x - y + 3z = 4

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

1. Step 1: Choose Equation 1 and solve for x: x = 1 - 2y + z
2. Step 2: Substitute the expression for x into Equations 2 and 3:
   • 2(1 - 2y + z) - y + 3z = 4
   • 3(1 - 2y + z) + y - 2z = 5
3. Step 3: Simplify and solve the new system:
   • -5y + 5z = 2
   • -5y + z = 2

   Solving this system (you can use elimination or substitution again), we find y = 0 and z = 2.

4. Step 4: Substitute y = 0 and z = 2 back into the equation x = 1 - 2y + z:
   x = 1 - 2(0) + 2 = 3
5. Step 5: Check the solution (x = 3, y = 0, z = 2) in all three original equations. You'll find that they are all satisfied.

Conclusion:

The substitution method is a powerful tool for solving systems of equations, even with three or more variables. By systematically eliminating variables and substituting expressions, you can find the unique solution to the system, if one exists.