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 + 3h
|Cost of Group B
10 + 2h
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
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
|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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