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
h

Cost of Group A
7 + 3h
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. :)

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