1. Solve linear systems by substitution.
7.4
Solve linear systems by substitution.

2. Solving Systems of Linear Equations by Substitution
7.4
Solving Systems of Linear Equations by Substitution
1. Solve linear systems by substitution.
2. Solve special systems by substitution.
3. Solve linear systems with fractions and decimals.

3. Solve linear systems by substitution.
Objective 1
Solve linear systems by substitution.

4. Definitions
Graphing to solve a system of equations has a serious drawback. For example, consider the system graphed below. It is difficult to determine an accurate solution of the system from the graph. As a result, there are algebraic methods for solving systems of equations. The substitution method, which gets its name from the fact that an expression in one variable is substituted for the other variable, is one such method.

5. Using the Substitution Method
Classroom Example 1
Using the Substitution Method
Solve the system by the substitution method.
We check that the ordered pair (8, –4) is the solution by substituting 8 for x and –4 for y in both equations.

6. Using the Substitution Method
Classroom Example 2
Using the Substitution Method
Solve the system by the substitution method.
Check that (15, –6) is the solution. Both results are true. The solution set is {(15, –6)}.

7. Solving a Linear System by Substitution
Step 1 Solve one equation for either variable. If one of the equations has a variable term with coefficient 1 or −1, choose it because the substitution method is usually easier.
Step 2 Substitute for that variable in the other equation. The result should be an equation with just one variable.
Step 3 Solve the equation from Step 2.
Step 4 Find the other value. Substitute the result from Step 3 into the equation from Step 1 and solve for the other variable.
Step 5 Check the values in both of the original equations. Then write the solution set as a set containing an ordered pair.

8. Using the Substitution Method
Classroom Example 3
Using the Substitution Method
Solve the system by the substitution method.
Solve equation (1) for x.
Check that (3, –1) is the solution. Both results are true. The solution set is {(3, –1)}.

9. Solve special systems by substitution.
Objective 2
Solve special systems by substitution.

10. Solving an Inconsistent System Using Substitution
Classroom Example 4
Solving an Inconsistent System Using Substitution
Solve the system by the substitution method.
The false result means that the equations in the system have graphs that are parallel lines. The system is inconsistent and has no solution, so the solution set is

11. Solving a System with Dependent Equations Using Substitution
Classroom Example 5
Solving a System with Dependent Equations Using Substitution
Solve the system by the substitution method.
Solve the first equation for x.
The true result means that every solution of one equation is also a solution of the other, so the system has an infinite number of solutions. The solution set is {(x, y) | x + 3y = –7}.

12. Solve linear systems with fractions and decimals.
Objective 3
Solve linear systems with fractions and decimals.

13. Using the Substitution Method (Fractional Coefficients)
Classroom Example 6
Using the Substitution Method (Fractional Coefficients)
Solve the system by the substitution method.
Clear the equations of fractions by multiplying by the LCD, 6 for the first equation and 2 for the second equation.

14. Using the Substitution Method (Fractional Coefficients) (cont.)
Classroom Example 6
Using the Substitution Method (Fractional Coefficients) (cont.)
Solve the second equation for x.
Check (0, –1) in both of the original equations. The solution set is {(0, –1)}.

15. Using the Substitution Method (Decimal Coefficients)
Classroom Example 7
Using the Substitution Method
(Decimal Coefficients)
Solve the system by the substitution method.
Clear the equations of decimals by multiplying both equations by 10.
Solve the second equation for 𝑥.

16. Classroom Example 7
Using the Substitution Method
(Decimal Coefficients) (cont.)
Substitute and solve.
Check (7, –2) in both of the original equations. The solution set is {(7, –2)}.