1) The document describes different geometric figures based on systems of equations in one, two, and three dimensions. 2) It provides examples of solving systems of equations in two and three variables, eliminating variables to solve the systems. 3) The solutions provided are the point (-2, 6, -3) for a system in three variables, and that some systems have infinite solutions or no solution.

1. What geometric figure is described by each of the following: In one dimension: x = 3 In two dimensions: 2x + 3y = 6 3) In three dimensions: 2x + 4y + z = 8 0 3 A plane

3. Solve the system of equations. 5x + 3y + 2z = 2 1 2 3 x 2 4x + 2y – 2z = 10 9x + 5y = 12 x 2 4x + 2y – 2z = 10 2 3 x + 4y + 2z = 16 5x + 6y = 26 Example 5-1a

4. x -6 -54x – 30y = -72 9x + 5y = 12 x 5 5x + 6y = 26 25x + 30y = 130 -29x = 58 -29 = -29 x = -2 5x + 6y = 26 5(-2) + 6y = 26 -10 + 6y = 26 6y = 36 y = 6

5. 1 2 3 x = -2 y = 6 z = -3 1 5(-2) + 3(6) + 2z = 2 -10 + 18 + 2z = 2 8 + 2z = 2 (-2, 6, -3) 2z = -6 z = -3

6. First equation Third equation Subtract to eliminate z. Notice that the z terms in each equation have been eliminated. The result istwo equations with the two same variables x and y. Example 5-1a

7. Add to eliminate y. Divide by 29. Equation with two variables Multiply by 5. Replace x with –2. Multiply. Simplify. Step 2Solve the system of two equations. Substitute –2 for x in one of the two equations with two variables and solve for y. Example 5-1a

8. Equation with three variables Replace x with –2 and y with 6. Multiply. Simplify. Step 3Substitute –2 for x and 6 for yin one of the original equations with three variables. Answer: The solution is (–2, 6, –3). You can check this solution in the other two original equations. Example 5-1a

9. Solve the system: 5x + 2y = 43x + 4y + 2z = 67x + 3y + 4z = 29 5x + 2y = 4 x 2 6x + 8y + 4z = 12 x -1 -7x – 3y – 4z = -29 -x + 5y = -17 -x + 5y = -17 x 5 -5x + 25y = -85 5x + 2y = 4 5x + 2y = 4 5x + 2(-3)= 4 27y = -81 5x – 6 = 4 y = -3 5x = 10 x = 2

10. 5x + 2y = 43x + 4y + 2z = 67x + 3y + 4z = 29 x = 2 y = -3 3(2) + 4(-3) + 2z = 6 6 – 12 + 2z = 6 – 6 + 2z = 6 2z = 12 z = 6 (2, -3, 6)

11. Solve the system of equations. x 3 x -2 -18x – 12y – 24z = 6 False No solution Eliminate x in the second two equations. Example 5-3a

12. Solve the system of equations. x 3 6x + 3y – 9z = 15 x -1 -6x – 3y +9z = -15 0 = 0 True Equations represent the same plane Is equation 2 the same plane? Eliminate y in the first and third equations. Example 5-2a

13. x -2 -4x – 2y + 6z = -10 x + 2y – 4z = 7 -3x + 2z = -3 Planes intersect in a line Infinite solutions Eliminate y in the first and second equations. Example 5-2a

14. Solve the system of equations. Answer:(–1, 2, –4) Example 5-1b

15. Solve the system of equations. Answer:There are an infinite number of solutions. Example 5-2b

16. Solve the system of equations. Answer:There is no solution of this system. Example 5-3b