# Math Help - Systems of Equations/Inequalities

1. ## Systems of Equations/Inequalities

1.) Solve the following system of equations using Gaussian elimination:
3w+10x-2y+3z=55
w+12x-11y-z= 69
-3w-6x-5y-10z=-47
-3w+9x+5y+4z=-1

Here's what I did so far:
swap R1 and R2 (row 1 and row 2)
-3R1+R2 (multiply row 1 by -3 and add row 2 to that product)
3R1+R3
3R1+R4

Now, after that I think I may have made a mistake because I eventually get some very odd fractions that could not possibly be correct. Can somebody tell me what the next steps are???

2.) Rachel has 3 investments for a total of $9000. They earn interest at 5%, 7%, and 9%, respectively. The total income is$7100. The income from the 9% investment exceeds the total from the other two investments by $1900. How much money is the investment that earns the 5% interest? I have absolutely no clue how to write the system for this. Can somebody at least help me get started??? I would really appreciate your help on this! Thank you!!! 2. Hello, $\text{I}\heartsuit\text{themusic29}$ ! I assume there is a typo in #2 . . . 2.) Rachel has 3 investments for a total of$90,000.
They earn interest at 5%, 7%, and 9%, respectively.
The total income is $7100. The income from the 9% investment exceeds the total from the other two investments by$1900.
How much money is the investment that earns the 5% interest?

Let . $\begin{array}{ccc}x &=& \text{amount at 5\%} \\ y &=& \text{amount at 7\%} \\ z &=& \text{amount at 9\%} \end{array}$

A total of $90,000 was invested: . $x + y + z \:=\:90,000\;\;{\color{blue}[1]}$ The total interest was$7100.
. . $0.05x + 0.07y + 0.09z \:=\:7100 \quad\Rightarrow\quad 5x + 7y + 9z \:=\:710,000\;\;{\color{blue}[2]}$

The interest from 9% is 1900 more than the total of the 5% and 7% investments.
. . $0.09z \:=\:0.05x + 0.07y + 1900\quad\Rightarrow\quad 5x + 7y - 9z \:=\:-190,000 \;\;{\color{blue}[3]}$

Subtract [3] from [2]: . $18z \:=\:900,000 \quad\Rightarrow\quad z \:=\:50,000$

$\begin{array}{ccccccccc}\text{Substitute into {\color{blue}[1]}:} & x + y + 50,000 &=&90,000 & \Rightarrow & x + y &=& 40,000 & {\color{blue}[3]}\\
\text{Substitute into {\color{blue}[2]}:} & 5x + 7y + 450,000 &=& 710,000 & \Rightarrow & 5x + 7y &=& 260,000 & {\color{blue}[4]}
\end{array}$

$\begin{array}{cccc}\text{Multiply {\color{blue}[3]} by 7:} & 7x + 7y &=& 280,000 \\ \text{Subtract {\color{blue}[4]}:} & 5x + 7y &=& 260,000 \end{array}$

And we get: . $2x \:=\:20,000\quad\Rightarrow\quad x \;=\;10,000$

Therefore, $10,000 was invested at 5%. 3. Originally Posted by iheartthemusic29 1.) Solve the following system of equations using Gaussian elimination: 3w+10x-2y+3z=55 w+12x-11y-z= 69 -3w-6x-5y-10z=-47 -3w+9x+5y+4z=-1 Here's what I did so far: swap R1 and R2 (row 1 and row 2) -3R1+R2 (multiply row 1 by -3 and add row 2 to that product) 3R1+R3 3R1+R4 Now, after that I think I may have made a mistake because I eventually get some very odd fractions that could not possibly be correct. Can somebody tell me what the next steps are??? 2.) Rachel has 3 investments for a total of$9000. They earn interest at 5%, 7%, and 9%, respectively. The total income is $7100. The income from the 9% investment exceeds the total from the other two investments by$1900. How much money is the investment that earns the 5% interest?

I have absolutely no clue how to write the system for this. Can somebody at least help me get started???

I would really appreciate your help on this! Thank you!!!
Swap r1 and R2
-3r1+r2
3r1+r3
3r1+r4
26/12r1+r2
-30/12r1+r3
These are the next few steps to number 1. Is this what you did?