Simultaneous Equations 3X3 - Questions
The solutions to the \(3 \times 3\) systems of simultaneous equations:
\(x + y + z = 6\,\,\,\,\,\,\,\,\,\,2x + 3y + z = 13\,\,\,\,\,\,\,\,\,\,x + 2y - z = 5{\rm{ }}\)
\begin{aligned}
x+y+z&=6 \cdots \cdots (1)\\
2 x+3y+z&=13\cdots \cdot \cdot (2)\\
x+2y-z&=5\cdots \cdots (3)\\
(1)+(3) \qquad 2x+3y&=11 \cdots \cdot \cdot(4)\\
(2)+(3) \qquad 3x+5y&=18 \cdots \cdot \cdot (5)\\
(4) \times 3 \qquad\;\;\, 6x+9y&=33 \cdots \cdot \cdot (6)\\
(5) \times 2 \qquad\, 6x+10y&=36 \cdots \cdot \cdot (7)\\
(6)-(7) \qquad \quad \;\, -y&=-3 \rightarrow y=3\\
\text { In }(4) \qquad\quad 2x+9&=11 \;\rightarrow x=1\\
\text { In } (1)\quad \;\;\; 1+3+z&=6 \;\;\;\rightarrow z=2
\end{aligned}
The solutions to the \(3 \times 3\) system of simultaneous equations:
\(2x + y - z = 9{\rm{ }}\) \(5x + 2z = - 3\) \(7x - 2y = 1\)
\begin{aligned}
2x+\;y\;-\,z&=9 \cdots \cdots (1)\\
5x+\quad+2z&=-3\cdots \cdot (2)\\
7x-2y\quad\;\;\; &=1\cdots \cdots (3)\\
(1)\times 2 \quad 4x+2y-2z&=18 \cdots \cdot \cdot(4)\\
(4)+(2) \qquad\;\, 9x+2y&=15 \cdots \cdot \cdot (5)\\
(3)+(5) \qquad \qquad\, 16x&=16 \;\rightarrow x=1\\
\text { In }(2) \qquad\qquad 5+2z&=-3 \rightarrow z=-4\\
\text { In } (1)\qquad \;\;\, 2+y+4&=9 \;\;\;\rightarrow y=3
\end{aligned}
The solutions to the \(3 \times 3\) system of simultaneous equations:
\(3x - y - 2z = - 15\) \({\rm{2}}x - 3y - 5z = - 15\) \({\rm{ 5}}x + 2y + 3z = - 16\)
\begin{align}
\begin{array}{l}
\\
\\
\\
(1) \times 2\\
(3)+(4)\\
(1)\times 3\\
(6) - (2)\\
(5)-(7)\\
\text {In } (7)\\
\text {In } (1)
\end{array}&\;
\begin{array}{r}
3x-y-2z=-15\cdots \cdot (1)\\
2x-3y-5z=-15\cdots \cdot (2)\\
5x+2y+3z=-16\cdots \cdot (3)\\
6x-2y-4z=-30 \cdots\cdot(4)\\
11x\qquad\; -z=-46\cdots \cdot (5)\\
9x-3y-6z=-45\cdots \cdot (6)\\
7x\qquad\;\;-z=-30\cdots \cdot(7)\\
4 x=-16 \rightarrow x=-4\\
-28-z=-30 \rightarrow z=2\;\;\;\\
-12-y-4=-15 \rightarrow y=-1
\end{array}
\end{align}
Alice, Betty, and Carol work in a shop. When Alice works \(2\) hours Betty works \(3\) hours and Carol, \(4\) hours. Their Combined earnings total \(\$ 242\). When Alice works \(4\) hours, Betty works \(2\) hours and Carol \(3\) hours, their combined total is \(\$ 252\). When Alice works \(2\) hours, Betty works \(5\) hours and Carol works \(2\) hours, Their combined earnings total is \(\$ 250\). The hourly rate of Betty is?
\begin{align}
&\text{Let Alice }=a, \text{ Betty}=b,\text{ Carol}=c\\
&\text{where }a, b, c \text{ stand for the hourly rate of pay.}\\
&\begin{aligned}
\begin{array}{l}
\\
\\
\\
(1) -(3)\\
\\
(1)\times 2\\
(2) - (5)\\
\\
\text {From } (4)\\
\text {In } (6)\\
\\
\\
\\
\\
\end{array}&\;
\begin{array}{r}
\therefore 2a+3b+4c=242\cdots \cdot (1)\\
4a+2b+3c=252\cdots \cdot (2)\\
2a+5b+2c=250\cdots \cdot (3)\\
-2b+2c=-8\qquad \quad \;\;\\
b-c=4\cdots \cdots \; (4)\\
4a+6b+8c=484\cdots \cdot (5)\\
-4b-5c=-232\qquad\;\;\\
4b+5c=232\cdots \cdot (6)\\
b=4+c\qquad \;\;\,\\
4(4+c)+5c=232\qquad\quad \;\\
16+4c+5c=232\qquad\quad \\
9c=216 \rightarrow c=24\quad\; \\
\therefore b=4 +24\rightarrow b=28\\
\therefore \text{Betty gets } \$28/\mathrm{h}\quad\;
\end{array}
\end{aligned}
\end{align}
A fruit store is selling bags of cherries. A small bag sells for \({\rm{\$ 3}}\), the medium size for \({\rm{\$ 5}}\) and the large bag for \({\rm{\$ 7}}\). A total of \({\rm{15}}\) bags were sold and the store made \({\rm{\$ 77}}\). Two more medium size bags than small bags were sold. How many large bags were sold?
\begin{align}
&\text { Let the numbers be small } \rightarrow x, \text { medium } \rightarrow y \text {, large } \rightarrow \text { z }\\
&\qquad \qquad\qquad \begin{aligned}
\therefore x+y+z=15 \cdots (1)&\\
3x+5y+7z=77 \cdots(2)&\\
y=x+2 \cdots (3)&
\end{aligned}\\
&\begin{aligned}
\text{Sub } (3) \text{ into } (1)\quad & x+x+2+z=15\\
&2x+z=13 \cdots \cdots(4)
\end{aligned}\\
&\begin{aligned}
\text{Sub } (3) \text{ into } (2)\quad & 3x+5(x+2)+7z=77\\ &8x+7z=67\cdots \cdot \cdot \, (5)
\end{aligned}\\
&(4) \times 4 \qquad \qquad \;\;\;8x+4z=52 \cdots\cdot \cdot (6)\\
&(5)-(6) \qquad \qquad \qquad \; 3z=15z \rightarrow 5\\
&\qquad \qquad \qquad \quad \therefore 5 \text{ large bags}
\end{align}