1. Chapter 6 Systems of Linear Equations
TMAT 103
Chapter 6
Systems of Linear Equations

2. §6.1 Solving a System of Two Linear Equations
TMAT 103
§6.1
Solving a System of Two Linear Equations

3. §6.1 – Solving a System of Two Linear Equations
Systems of two linear equations:

4. §6.1 – Solving a System of Two Linear Equations
Graphs of linear systems of equations with two variables
The two lines may intersect at a common, single point. This point, in ordered pair form(x, y), is the solution of the system
independent and consistent
The two lines may be parallel with no points in common; hence, the system has no solution
inconsistent
The two lines may coincide; the solution of the system is the set of all points on the common line
dependent

5. §6.1 – Solving a System of Two Linear Equations
Methods available to solve systems of equations
Addition-subtraction method
Method of substitution

6. §6.1 – Solving a System of Two Linear Equations
Solving a pair of linear equations by the addition-subtraction method
If necessary, multiply each side of one or both equations by some number so that the numerical coefficients of one of the variables are of equal absolute value.
If these coefficients of equal absolute value have like signs, subtract one equation from the other. If they have unlike signs, add the equations.
Solve the resulting equation for the remaining variable.
Substitute the solution for the variable found in step 3 in either of the original equations, and solve this equation for the second variable.
Check

7. §6.1 – Solving a System of Two Linear Equations
Examples – solve the following using the addition-subtraction method

8. §6.1 – Solving a System of Two Linear Equations
Solving a pair of linear equations by the method of substitution
From either of the two given equations, solve for one variable in terms of the other.
Substitute this result from step 1 in the other equation. Note that this step eliminates one variable.
Solve the equation obtained from step 2 for the remaining variable.
From the equation obtained in step 1, substitute the solution for the variable found in step 3, and solve this resulting equation for the second variable.
Check

9. §6.1 – Solving a System of Two Linear Equations
Examples – solve the following using the method of substitution

10. §6.1 – Solving a System of Two Linear Equations
Steps for problem solving
Read the problem carefully at least two times.
If possible, draw a picture or diagram.
Write what facts are given and what unknown quantities are to be found.
Choose a symbol to represent each quantity to be found.
Write appropriate equations relating these variables from the information given in the problem (there should be one equation for each unknown).
Solve for the unknown variables using an appropriate method.
Check your solution in the original equation.
Check your solution in the original verbal problem.

11. §6.1 – Solving a System of Two Linear Equations
Examples – solve the following
A plane can travel 900 mile with the wind in 3 hours. It makes the return trip in 3.5 hours. Find the rate of windspeed, and the speed of the plane
A chemist has a 5% solution and an 11% solution of acid. How much of each must be mixed to get 1000L of a 7% solution?

12. §6.2 Other systems of equations
TMAT 103
§6.2
Other systems of equations

13. §6.2 – Other systems of equations
Other types of problems can be solved using either the addition-subtraction method, or the method of substitution
Literal equations
coefficients are letters
will not be covered in this class
Non-linear equations
variables in denominator

14. §6.2 – Other systems of equations
Examples – solve the following using the method of substitution or addition-subtraction method

15. §6.3 Solving a System of Three Linear Equations
TMAT 103
§6.3
Solving a System of Three Linear Equations

16. §6.3 – Solving a System of Three Linear Equations
Systems of three linear equations:

17. §6.3 – Solving a System of Three Linear Equations
Graphs of linear systems of equations with three variables
The three planes may intersect at a common, single point. This point, in ordered triple form (x, y, z), is then the solution of the system.
The three planes may intersect along a common line. The infinite set of points that satisfy the equation of the line is the solution of the system.
The three planes may not have any points in common; the system has no solution. For example, the planes may be parallel, or they may intersect triangularly with no points in common to all three planes.
The three planes may coincide; the solution of the system is the set of all points in the common plane.

18. §6.3 – Solving a System of Three Linear Equations
Solving a pair of linear equations by the addition-subtraction method
Eliminate a variable from any pair of equations using the same technique from section 6.1
Eliminate the same variable from any other pair of equations.
The results of steps 1 and 2 is a pair of linear equations in two unknowns. Solve this pair for the two variables
Solve for the third variable by substituting the results from step 3 in any one of the original equations
Check

19. §6.3 – Solving a System of Three Linear Equations
Examples – solve the following using the addition-subtraction method

20. §6.3 – Solving a System of Three Linear Equations
Example – solve the following
75 acres of land were purchased for $142,500. The land facing the highway cost $2700/acre. The land facing the railroad cost $2200/acre, and the remainder cost $1450/acre. There were 5 acres more facing the railroad than the highway. How much land was sold at each price?