Pythagoras

==​Pythagoras ​ Solving for Systems of Equations! (oooh fancy) ==

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FIRST: How Do I Solve? Solving systems of equations graphically is considered by most to be the easiest way to solve systems of equations, though there are a few other options you have. But how do you even solve the problems, you ask? And what are the different ways you can solve them? =====

There are three different ways, or methods, to solve for systems of equations.


 * 1) The ELIMINATION (or ADDITION) Method.
 * 2) The GRAPHING Method
 * 3) The SUBSTITUTION Method.

When using the Elimination Method...

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Video not doing the job? No problem. Here's more on the Elimination Method straight from a reliable source: your 2009/2010 Algebra 1 textbook used by the fabulous Mrs. Wilson herself. If you happen to have a copy of the textbook and are thinking to yourself, "Self, what page would I find this on?" or, "Self, what chapter would I find this on?" we have the answer. It's 6-3, page 287. You're welcome.

The Addition and Subtraction Properties of Equality can be shown like this respectively (that means 'in that order') :

If a = b and c = d, then a + c = b + d. If a = b and c = d, then a - c = b - d.

You can use the properties of equality to solve a system using the wonderful elimination method. By adding or subtracting the variable, you can solve and //eliminate// the variable (huh, I wonder if that's why it's called the elimination method!)

When using the graphing method...

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THIS video not doing the job? Old mathematicians that you find strange to see on YouTube disconcerting? Ahh. In that case, here's some straight from a reliable source: your 2009/2010 Algebra 1 textbook that the wonderful Mrs. Wilson uses herself. If you're thinking to yourself, "Self, where would I find this in my textbook?" or, "Self, what chapter is this in?" here's your answer: it's 6-1, on page 276. You're welcome.

One way to solve LINEar equations (notice the LINE in linear!) is by graphing the equation (to form a LINE!) Look for any point common to all the lines. Any ordered pair in a system that makes all the equations affirmative is a solution to the system of equations. Here's a lovely picture:



The point's coordinates (1, 3) would be the solution to the set above.

Do you want to try a **//word problem//** ? Okay!

Suppose you are testing two fertilizers on bamboo plants A and B, which are growing under identical conditions. Plant A is 6 cm tall and growing at a rate of 4 cm/day. Plant B is 10 cm tall and growing at a rate of 2 cm/day. Write a system of equations that models the height of each plant H(d) as a function of days d.

WRITE: Plant A: H(d)= 6+4d Plant B: H(d)= 10 + 2d

The system is: H(d)=4d+6 H(d)=2d+10

Ready to solve? H(d)=4d+6 H(d)=2d+10

Since each equation is equal to H(d), the equations are equal to each other. 4d+6=2d+10

4d+6=2d+10 Subtract 6 from both sides to get the variable (d) alone.

4d+6-6=2d+10-6 4d=2d+4 Now, subtract 2d from both sides to get the variable alone.

4d-2d=2d-2d+4 2d=4 Last, divide both sides by 2 to get the variable alone.

2d/2=4/2

So, d=4/2. So d=2. Now, let's check the work!

H(d)=4d+6 H(d)=2d+10

H(2)=4(2)+6 H(2)=2(2)+10

4(2)+6=2(2)+10 8+6=4+10 14=14 Great work! Any questions? Look in the textbook at Systems of Equations, and look at the examples. :)

=__ Examples for Solving Systems of Equations __=

==== Your school must transport 193 people to a competition. There are 8 drivers available and two types of vehicles. The school buses seat 51 people each, and the minivans seat 8 people each. How many buses and minivans will be needed? ====

__ m=5 __
Three school buses and five minivans will be needed to transport 193 people.

Adding Equations:
5x-6y=-32

x=2
3x+6y=48

2x+5y=-22 10x+3=22

5(2x+5y=-22) 10x+25y=-110 __10x+3y=22__ 0+22y=-132 22y=-132 __y=-6__

2x+5y=-22 2x+5(-6)=-22 2x-30=-22 2x=8 __x=4__

(4,-6)

__ x=1 __
4x+2y=14 4(1)+2y=14 2y=10

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Suppose your class sells gift wrap for $4 per package and greeting cards for $10 per package. Your class sells 205 packages in all and receives a total of $1084. Find the number of each type of package sold. ======