**Algebra Review - Systems of Equations**

**1. Comparing Expressions and Graphs**

When you need to get your computer fixed or house cleaned, you encounter various rates and costs. Some say something like $10 plus $2 per hour or $5 plus $3 per hour. Comparing expressions with algebra helps see which one saves the most money depending on the amount of time you expect it to take.

Example: Two companies offer to clean your
house. Their rates are shown below:

Group A - $7 plus $3 per hour

Group B - $10 plus $2 per hour

Then make a similar table below - plotting various hours.

Number of hours |
Cost of Group A 7 + 3 h |
Cost of Group B 10 + 2h |

1 | 10 | 12 |

2 | 13 | 14 |

3 | 16 | 16 |

4 | 19 | 18 |

5 | 22 | 20 |

Now look at the table. If you need to clean for 2 hours, Group A would be cheaper. However, if you need to clean 3 hours, both Group A and Group B offer the same value. However, if you need to clean 4 hours or more, Group B would cost the least.

You can graph both equations on the same graph to compare, also. I do not have a graph available, but there should be two lines with different slopes. Where they intersect will be where they cost the same. If a line is below another, that line is cheaper, and so on.

**2. Finding Points of Intersection**

Find the intersection points of the following
equations:

y = -2x + 3

y = .5x - 2

First graph it: | |

You can see they intersect at (2,-1). That's basically how you use this method. To be sure your intersection point is correct, plug the point back into the equations and see if they come out equal. |

If the slopes are parallel (the coefficients of
the x variables), then the lines will **never** intersect.

If you have something like x + 2y = 6 and 3x +
6y = 18, notice you can simplify the second equation by dividing three on both
sides to get x + 2y = 6. Since both equations are the same, the lines will have
an **infinite** number of intersections since they lie on top of each other.

**3. Simultaneous Equations with Substitution**

__ Direct Substitution__ - This is
directly substituting a number into an equation to solve for the unknown.

y = 5 + 4x

y = 17

17 = 5 + 4x | Substitute y = 17 into the first equation. |

12 = 4x | Simplify and solve. |

x = 3 |

__ Indirect Substitution__ - This is
rewriting the equation so you can plug it in and solve for two variables.

x + 2y = 6

2x + 3x = 8

x + 2y = 6 | Pick the simpler looking equation and solve for any variable; here we choose to solve for x. |

x = -2y + 6 | |

2(-2y + 6) + 3y = 8 | Now plug that into the second equation. |

-4y + 12 + 3y = 8 | Simplify. |

-y = -4 y = 4 |
Solve. |

x + 2(4) = 6 x = 8 |
Plug the y into one of the equations to solve for x. |

Now you have x = -2 and y = 4 |

__ Elimination__
- This is when you eliminate one variable by adding the equations together and
solving for the remaining unknown.

a + b = 8

a - b = 4

a + b = 8 a - b = 4 |
See how adding +b and -b cancels them out? |

2a = 12 | Simplify and solve. |

a = 6 |

You can always modify equations by multiplying/adding/etc. on both sides in order to eliminate one variable.

2a + b = 8

a - 2b = 4

2a + b = 8 a - 2b = 10 |
You can multiply the first equation by 2. |

4a + 2b = 16 a - 2b = 4 |
Hey! Now you can eliminate 2b! |

5a = 20 a = 4 |
Solve and you are done. |

**4. Rate and Mixture Problems**

If you want this info, tell me and I will work on it. :)