Systems of linear equations in three variables ppt

1. Systems of Linear Equations
in Three Variables

2. Objective: Students solve systems of linear
equations and inequalities in three variables by
substitution, with graphs.

3. A linear equation in three
variables x, y, and z is an
equation of the form ax +
by + cz = d, where a, b, and
c are not all zero.
The following is an
example of a system of
three linear equations in
three variables:
2x + y – z = 5
3x – 2y + z = 16
4x + 3y – 5z = 3

4. Solve Systems of Linear Equations in Three Variables

5. Example 1: Solve the system using the Linear Combination Method.

6. Step 1: Eliminate one of the variables in two of the original equations.

7. Step 1: Eliminate one of the variables in two of the original equations.

8. Step 2: Solve the new system of equations

9. Then, Substitute z = 4 into new equation 1 or 2 and
solve for x

10. After that, step 3: Substitute x = -3 and z = 4 into any of the
original equations and solve for y.

13. Since 0 ≠ 8 the result is a false equation. This system has NO SOLUTIONS.

14. Try.

15. Solve the system :
X + y + 2z = 9
2x+ 4y -3z =1
3x+ 6y -5z =0
Then find the value of x + y + z

16. solving a word problem with 3 unknowns using a linear
equation
Example.
Amanda, Henry, and Scott have a total of $89 in
thier wallets. Amanda has 6$ less than Scott. Henry
has 3 times what scott has. How much does each
have?

17. Answer
Let x be the amount of money Amanda has
Let y be the amount of money Henry has
Let z be the amount of money Scott has
Amanda, Henry, and Scott have a total of $89 in their wallets
The above statement gives the following equation
x + y + z = 89
Amanda has 6$ less than Scott
The above statement gives the following equation
x = z - 6
Henry has 3 times what scott has.
The above statement gives the following equation
y = 3z

18. Continue...
Replace x = z - 6 and y = 3z in equation 1
z - 6 + 3z + z = 89
5z - 6 = 89
5z - 6 + 6 = 89 + 6
5z = 95
z = 19
y = 3z = 3 × 19 = 57
13 = x
So, Scott has 19 dollars, Henry has 57 dollars, and Amanda has 13 dollars.