Section 4.2 Solving Systems of Equations by Substitution

Objectives Solve systems of linear equations by substitution Find a substitution equation Solve systems of linear equations that contain fractions Use substitution to identify inconsistent systems and dependent equations

Objective 1: Solve Systems of Linear Equations by Substitution When solving a system of equations by graphing, it is often difficult to determine the coordinates of the intersection point.  For example, a solution of (7/8, 3/5) would be almost impossible to identify accurately.  In this section, we will discuss a second, more precise method for solving systems that does not involve graphing.

Objective 1: Solve Systems of Linear Equations by Substitution One algebraic method for solving a system of equations is the substitution method. It is introduced in the following example. The substitution method works well for solving systems where one equation is solved, or can be easily solved, for one of the variables. To solve a system of equations in x and y by the substitution method, follow these steps. The Substitution Method: 1. Solve one of the equations for either x or y. If this is already done, go to step 2. (We call this equation the substitution equation.) 2. Substitute the expression for x or for y obtained in step 1 into the other equation and solve that equation. 3. Substitute the value of the variable found in step 2 into the substitution equation to find the value of the remaining variable. 4. Check the proposed solution in each equation of the original system. Write the solution as an ordered pair.

EXAMPLE 2 Solve the system:

Objective 2: Find a Substitution Equation Sometimes neither equation of a system is solved for a variable.  In such cases, we can find a substitution equation by solving one of the equations for one of its variables.

EXAMPLE 3 Solve the system:

Objective 3: Solve Systems of Linear Equations that Contain Fractions It is usually helpful to clear any equations of fractions and combine any like terms before performing a substitution.

EXAMPLE 5 Solve the system:

Objective 4: Use Substitution to Identify Inconsistent Systems and Dependent Equations In the previous section, we solved inconsistent systems and systems of dependent equations graphically. We can also solve these systems using the substitution method.

EXAMPLE 6 Solve the system: