1. 6.2 Using Substitution to Solve Systems

2. Example: Making a Substitution for y
Solve the system
Equation (1)
Equation (2)

3. Solution
From equation (1). We know that for any solution of the system, the value of y is equal to the value of x So, we substitute x + 3 for y in equation (2):
Equation (2)

4. Solution
Note that, by making this substitution, we now have an equation in one variable. Next, we solve that equation for x:

5. Solution
Thus the x-coordinate of the solution is 1. To find the y-coordinate, we substitute 1 for x in either of the original equations and solve for y:
Equation (1)

6. Solution
So, the solution is (1, 4). We can check that (1, 4) satisfies both of the system’s equations:

7. Solution
Or, we can verify that (1, 4) is the solution by graphing equations (1) and (2) and checking that (1, 4) is the intersection point of the two lines. To do so on a graphing calculator, we must first solve equation (2) for y:

8. Solution

9. Using Substitution to Solve a Linear System
To use substitution to solve a system of two linear
equations,
1. Isolate a variable on one side of either equation.
2. Substitute the expression for the variable found in step 1 into the other equation.
3. Solve the equation in one variable found in step 2.
4. Substitute the solution found in step 3 into one of the original equations, and solve for the other variable.

10. Example: Isolating a Variable and Then Using Substitution
Solve the system
Equation (1)
Equation (2)

11. Solution
We begin by solving for one of the variables in one of the equations. We can avoid fractions by choosing to solve equation (1) for y:
Equation (1)

12. Solution
Next, we substitute –3x – 7 for y in equation (2) and solve for x:
Equation (2)

13. Solution
Finally, we substitute –2 for x in the equation y = –3x – 7 and solve for y:
The solution is (–2, –1). We could then verify our work by checking that (–2, –1) satisfies both of the original equations.

14. Example: Applying Substitution to an Inconsistent System
Consider the linear system
Equation (1)
Equation (2)
The graphs of the equations are parallel lines (why?), so the system is inconsistent and the solution is the empty set. What happens when we solve this system by substitution?

15. Solution We substitute 3x + 2 for y in equation (2) and solve for x:
false
We get the false statement 2 = 4.

16. Inconsistent System of Equations
If the result of applying substitution to a system of equations is a false statement, then the system is inconsistent; that is, the solution set is the empty set.

17. Example: Applying Substitution to an Dependent System
In an example in Section 6.1, we found that the system
Equation (1)
Equation (2)
is dependent and that the solution set is the infinite set of solutions of the equation y = 3x – 5. What happens when we solve this system by substitution?

18. Solution We substitute 3x – 5 for y in equation (2) and solve for x:
true
We get the true statement 10 = 10.

19. Dependent System of Equations
If the result of applying substitution to a system of equations is a true statement that can be put into the form a = a, then the system is dependent; that is, the solution set is the set of ordered pairs represented by every point on the (same) line.