Systems of Linear Equations and Inequalities

Solving Systems of Linear Equations by the Substitution Method

Systems of linear equations consist of two or more equations with the same set of variables. The substitution method is a systematic approach for finding the solution to such systems, which may have one solution, no solution, or infinitely many solutions.

• Objective 1: Solve linear systems by the substitution method.

• Objective 2: Use the substitution method to identify systems with no solution or infinitely many solutions.

• Objective 3: Solve application problems using the substitution method.

Steps for Solving Linear Systems by Substitution

1. Solve one equation for one variable in terms of the other variable. If an equation is already solved for a variable, this step can be skipped.

2. Substitute the expression from step 1 into the other equation. This results in an equation with one variable.

3. Solve the resulting equation for the single variable.

4. Back-substitute the value found into the equation from step 1 to find the value of the remaining variable.

5. Check the proposed solution in both original equations to verify correctness.

Example 1: Solving a System by Substitution

Suppose we have the system:

• Equation 1:

• Equation 2:

Step 1: The first equation is already solved for . Substitute into the second equation:

Step 2: Simplify and solve for :

Step 3: Back-substitute into the first equation:

Step 4: The solution is .

Step 5: Check in both equations:

• Equation 1: (True)

• Equation 2: (True)

The solution set is .

Example 2: Solving a System by Substitution (Variable Solved in Second Equation)

Suppose we have:

• Equation 1:

• Equation 2:

Step 1: Equation 2 is already solved for . Substitute into Equation 1:

Step 2: Simplify and solve for :

Step 3: Back-substitute into Equation 2:

Step 4: The solution is .

Step 5: Check in both equations:

• Equation 1: (True)

• Equation 2: (True)

The solution set is .

Identifying Special Types of Systems

No Solution (Inconsistent System)

If, after substitution, both variables are eliminated and a false statement (such as ) results, the system is inconsistent and has no solution.

Example: After substitution, if you obtain , the solution set is (the empty set).

Infinitely Many Solutions (Dependent System)

If, after substitution, both variables are eliminated and a true statement (such as ) results, the system is dependent and has infinitely many solutions.

Example: After substitution, if you obtain , the solution set contains infinitely many solutions (all points that satisfy the equations).

Applications: Supply and Demand Models

Systems of equations are used to model real-world scenarios such as supply and demand. In these models, the price and quantity of goods are related by two equations: one for demand and one for supply. The equilibrium point is where the two equations intersect.

• Demand equation: Relates price to the quantity demanded.

• Supply equation: Relates price to the quantity supplied.

Example: Suppose the demand and supply for two-bedroom apartments are modeled by:

• Demand:

• Supply:

To find the equilibrium, set the two equations equal:

(in thousands of apartments)

Back-substitute to find :

The equilibrium is , meaning when rents are $900 per month, consumers will demand 30,000 apartments and suppliers will offer 30,000 apartments for rent.

Additional info: The included image is the cover of the referenced textbook, which may help students identify the source but is not directly relevant to the substitution method. It is included here only if the instructor wishes to visually connect the notes to the course text.