Substitution Method Study Guide

Tessa Steinberg

Topic: Substitution Method in Solving Systems of Equations

Overview

• The substitution method is a technique used in algebra to solve systems of equations. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Equations Provided

1. Equation 1:
   2x - 3y = -1

2. Equation 2:
   y = x - 1

Steps for Solving the System

1. Identify the equations:

   • First, we have the equation 2x - 3y = -1 and the substitution for y given by y = x - 1.

2. Substitute the value of y into the first equation:

   • Replace y in the first equation with the expression from the second equation:

   2x - 3(x - 1) = -1

3. Simplify the equation:

   • Distributing the -3 across the terms inside the parentheses:
     2x - 3x + 3 = -1

   • Combine like terms:
     -x + 3 = -1

4. Solve for x:

   • To isolate x, subtract 3 from both sides:
     -x = -1 - 3

   • This simplifies to:
     -x = -4

   • Multiply both sides by -1 to solve for x:
     x = 4

5. Substitute x back into the equation for y:

   • Substitute x = 4 back into the second equation (y = x - 1):
     y = 4 - 1

   • Therefore:
     y = 3

Conclusion

• The solution to the system of equations is:
  x = 4 and y = 3

Summary of the Substitution Method

• The substitution method involves:

  • Selecting an equation to solve for one variable.

  • Replacing that variable in the other equation.

  • Solving the resulting equation for that variable.

  • Substituting back to find the other variable.

• The substitution method is especially effective when one equation is already solved for one variable, as in this case where y was isolated to find its value first.

Important Notes

• The method can be applied to different types of equations, including linear and nonlinear equations.

• It is essential to check the solution by substituting both values back into the original equations to ensure they satisfy both equations.