Thread: solving with 3 variables?!

1. solving with 3 variables?!

Word problem: solving with 3 variables:

Lila has three investments totaling $100,000. These investments earn interest at 7%, 9%, and 11% respectively. Lila's total income from these investments is$9800. The income from the 11% investment exceeds the total income from the other two investments by $1200. Find how much Lila has invested at 11%. Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 40 cents each, oatmeal muffins at 45 cents each, and cranberry muffins at 50 cents each. Charles buys some of each kind and chooses twice as many cranberry muffins as chocolate chip muffins. If he spends$6.45 on 14 muffins, how many chocolate chip muffins did he buy?

Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13

Solve the following system. What is the value of y?
-5x + 2z = 18
x + 4y + z = -25
-3x + 5y - z = -12

Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13

2. Originally Posted by xbchgrl04x
Word problem: solving with 3 variables:

Lila has three investments totaling $100,000. These investments earn interest at 7%, 9%, and 11% respectively. Lila's total income from these investments is$9800. The income from the 11% investment exceeds the total income from the other two investments by $1200. Find how much Lila has invested at 11%. ... Let a, b and c denote the amounts of the three investments. Then you know:$\displaystyle \left|\begin{array}{rcl}a+b+c&=&100,000 \\ 0.07a+0.09b+0.11c&=&9800 \\ 0.07a+0.09b+1200&=&0.11c\end{array}\right.$Choose the method to solve this system which you know best. You should get: (a; b; c) = (10,000 ; 40,000 ; 50,000) 3. Originally Posted by xbchgrl04x ... Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 40 cents each, oatmeal muffins at 45 cents each, and cranberry muffins at 50 cents each. Charles buys some of each kind and chooses twice as many cranberry muffins as chocolate chip muffins. If he spends$6.45 on 14 muffins, how many chocolate chip muffins did he buy?

...
Let c denote the numbers of chocolate muffins, o the numbers of oatmeal muffins and b the numbers of cranberry muffins.

Then you know:

$\displaystyle \left|\begin{array}{rcl}c+o+b&=&14 \\ 0.4c+0.45o+0.5b&=&6.45\\ b = 2c\end{array}\right.$

Choose the method to solve this system which you know best. You should get:
(c; o; b) = ( 3; 5; 6)

4. Originally Posted by xbchgrl04x
...

Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13

Solve the following system. What is the value of y?
-5x + 2z = 18
x + 4y + z = -25
-3x + 5y - z = -12

Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13

These 2 (two) systems can be solved using one of the standard methods. The results are triples of integer numbers. Nothing very tricky!

So I need to know where you have difficulties to do them. Otherwise I can't help you.

5. Originally Posted by earboth
These 2 (two) systems can be solved using one of the standard methods. The results are triples of integer numbers. Nothing very tricky!

So I need to know where you have difficulties to do them. Otherwise I can't help you.
Honestly, I get stuck from the begining...

I have tried looking up examples online and this is what I get:

-5x + 2z = 18
x + 4y + z = -25
-3x + 5y - z = -12

-5x + 2z = 18
x + 4y + z = -25
-4x + 4y +32 = -7

x + 4y + z = -25
-3x + 5y - z = -12
-2x + 9y = -37

-4x + 4y + 3z = -7
-2x + 9y = -37

and then I am lost..

6. Originally Posted by xbchgrl04x
Honestly, I get stuck from the begining...

...
Isn't the good ol' Google working anymore?

Have a look here: System of linear equations - Wikipedia, the free encyclopedia

7. Originally Posted by earboth
Isn't the good ol' Google working anymore?

Have a look here: System of linear equations - Wikipedia, the free encyclopedia
Okay, I tried them over again.. can you at least let me know if I got these examples right?

Solve the following system. What is the value of y?
4x - 5z = -33
x + y + 2z = 11
-x + 3y - z = 6
y = 5

Solve the following system. What is the value of y?
-5x + 2z = 18
x + 4y + z = -25
-3x + 5y - z = -12
y= -1

Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13
y= 3

8. Originally Posted by xbchgrl04x
Okay, I tried them over again.. can you at least let me know if I got these examples right?

...
All three examples have triples of integer numbers as solution.

My results differ from your's in all three cases.

So I suggest that you do one example completely (not only publishing one value) so I can see where you need some additional assistance.

9. Originally Posted by xbchgrl04x
Solve the following system. What is the value of y?
-5x - z = 2
x + 4y - 2z = 1
2x - 4y - z = -13
I'll show you this one -- you try the others.

I'm going to change the system into "triangular form," which looks something like this:
\displaystyle \begin{aligned} x&\; +& ?y&\; +& ?z&\; = ? \\ &\; {\color{white}.}& y&\; +& ?z&\; = ? \\ &\; {\color{white}.}& &\; {\color{white}.}& z&\; = ? \end{aligned}
...where "?" are constants, and any of the "+"s could be minus signs.

(To be continued below)

01

\displaystyle \begin{aligned} -5x&\; {\color{white}.}& {\color{white}.}&\; -& z&\; = 2 \\ x&\; +& 4y&\; -& 2z&\; = 1 \\ 2x&\; -& 4y&\; -& z&\; = -13 \end{aligned}

Switch the position of Eq1 & Eq2:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ -5x&\; {\color{white}.}& {\color{white}.}&\; -& z&\; = 2 \\ 2x&\; -& 4y&\; -& z&\; = -13 \end{aligned}

Multiply Eq1 by 5 and add the result to Eq2. Put the result in the Eq2 slot:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ {\color{white}.}&\; {\color{white}.}& 20y&\; -& 11z&\; = 7 \\ 2x&\; -& 4y&\; -& z&\; = -13 \end{aligned}

Divide Eq2 by 20:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ {\color{white}.}&\; {\color{white}.}& y&\; -& 0.55z&\; = 0.35 \\ 2x&\; -& 4y&\; -& z&\; = -13 \end{aligned}

Multiply Eq1 by -2 and add the result to Eq3. Put the result in the Eq3 slot:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ {\color{white}.}&\; {\color{white}.}& y&\; -& 0.55z&\; = 0.35 \\ {\color{white}.}&\; {\color{white}.}& -12y&\; +& 3z&\; = -15 \end{aligned}

Multiply Eq2 by 12 and add the result to Eq3. Put the result in the Eq3 slot:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ {\color{white}.}&\; {\color{white}.}& y&\; -& 0.55z&\; = 0.35 \\ {\color{white}.}&\; {\color{white}.}& {\color{white}.}&\; {\color{white}.}& -3.6z&\; = -10.8 \end{aligned}

Divide Eq3 by -3.6:
\displaystyle \begin{aligned} x&\; +& 4y&\; -& 2z&\; = 1 \\ {\color{white}.}&\; {\color{white}.}& y&\; -& 0.55z&\; = 0.35 \\ {\color{white}.}&\; {\color{white}.}& {\color{white}.}&\; {\color{white}.}& z&\; = 3 \end{aligned}

You can see how easy it is from here to find x and y. I'll let you do that. (If you can see the white dots in the equations above, ignore them. I only put them there to line things up.)

01

11. When I think of systems of equations with three variables, I usually just use the Gaussian Method of Elimination...which is right here: Systems of Linear Equations: Gaussian Elimination