Review of Chapter 8: Solving A System of Linear Equations

From class, we had the problem: 3x + (1/3)y = 10; 2x - (5/3)y = 35. So, multiply 3 to everything. So, 3(3x + (1/3)y = 10); 3(2x - (5/3)y = 35). Therefore, you have 9x + y = 30; 6x - 5y = 105, respectively. Now, multiply the first linear equation by 5:5(9x + 5 = 30). So, 45x + 5y = 150; 6x - 5y = 105. Add them, and you get, 51x = 255. Therefore, x = 5. Using the second equation, plug in 5 for x. So, 3(5)+(1/3)y = 10. Therefore, 15 + (1/3)y = 10. Solve for y. So, (1/3)y = -5. Then, multiply each side by 3. So y = -15. QED

Carbon Dating - Exponential Relationship

AI tried to show a Star Trek clip, but couldn't get it. So here is a clip from Numb3rs about carbon dating.

Solving a Three Variable System of Equations

Here is the problem from class: a + b + c = 5, 2a + 3b + c = 10, 4a - b + c = 9. So, solve for c in the first equation. c = 5 - a - b. So, substitute the equation of c into the other equations. 2a + 3b + 5 - a - b = 10, 4a - b + 5 - a - b = 9. So, simplify the equations. a + 2b = 5, 3a - 2b = 4. So through elimination, add the equations. 4a = 9, therefore a = 2.5. So let's use two equations: a + b + c = 5, 4a - b + c = 9. So through elimination, add the equations. 5a + 2c = 14, and since a = 2.5, 5(2.5) + 2c = 14, 12.5 + 2c = 14, 2c = 1.5 and therefore, c = .75. So, a + b + c = 5 with the values of a and c, look like: 2.5 + b + .75 = 5. Solve for b. b = 1.75. Check your work and the values add up to 5.