3.1 Systems of Linear Equations in Three Variables

3.1 Systems of Linear Equations in Three Variables - Set 1

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Given the simultaneous linear equations:

\(\begin{aligned} 4x-2y+z&=2\quad &...\boxed{1} \\\\ 6x+7y-z&=3 \quad &...\boxed{2} \\\\ 5x+y+2z&=7 \quad &...\boxed{3} \end{aligned} \)

Find the value of \(x,\,y\) and \(z\).

(A)		\(\begin{aligned} x&=\dfrac{2}{13} \\\\ y&=\dfrac{9}{13} \\\\ z&=\dfrac{36}{13} \end{aligned}\)

(B)		\(\begin{aligned} x&=\dfrac{1}{13} \\\\ y&=\dfrac{8}{13} \\\\ z&=\dfrac{35}{13} \end{aligned}\)

(C)		\(\begin{aligned} x&=\dfrac{2}{13} \\\\ y&=\dfrac{9}{13} \\\\ z&=-\dfrac{36}{13} \end{aligned}\)

(D)		\(\begin{aligned} x&=\dfrac{1}{13} \\\\ y&=\dfrac{8}{13} \\\\ z&=-\dfrac{35}{13} \end{aligned}\)

Difficulty Level: Medium

3.1 Systems of Linear Equations in Three Variables - Set 1

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Solve the following simultaneous linear equations:

\(\begin{aligned} 3x-y+z&=2\quad &...\boxed{1} \\\\ 2x+3y-z&=1 \quad &...\boxed{2} \\\\ x+y+2z&=4 \quad &...\boxed{3} \end{aligned} \)

(A)		\(\begin{aligned} x&=\dfrac{9}{25} \\\\ y&=\dfrac{3}{5} \\\\ z&=-\dfrac{38}{25} \end{aligned}\)

(B)		\(\begin{aligned} x&=\dfrac{9}{25} \\\\ y&=-\dfrac{3}{5} \\\\ z&=\dfrac{38}{25} \end{aligned}\)

(C)		\(\begin{aligned} x&=\dfrac{9}{25} \\\\ y&=\dfrac{3}{5} \\\\ z&=\dfrac{38}{25} \end{aligned}\)

(D)		\(\begin{aligned} x&=-\dfrac{9}{25} \\\\ y&=\dfrac{3}{5} \\\\ z&=\dfrac{38}{25} \end{aligned}\)

Difficulty Level: Medium

3.1 Systems of Linear Equations in Three Variables - Set 1

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The expression \(px^2+qx+r\) has values of \(7,9,\) and \(30\) when \(x=0,1\) and \(-2\) respectively.

Find the values of \(p,q\) and \(r\).

(A)		\(\begin{aligned} p&=\dfrac{9}{5} \\\\ q&=\dfrac{5}{2} \\\\ r&=7 \end{aligned}\)

(B)		\(\begin{aligned} p&=\dfrac{9}{2} \\\\ q&=\dfrac{5}{2} \\\\ r&=-6 \end{aligned}\)

(C)		\(\begin{aligned} p&=\dfrac{9}{2} \\\\ q&=-\dfrac{5}{3} \\\\ r&=-7 \end{aligned}\)

(D)		\(\begin{aligned} p&=\dfrac{9}{2} \\\\ q&=-\dfrac{5}{2} \\\\ r&=7 \end{aligned}\)

Difficulty Level: Hard

3.1 Systems of Linear Equations in Three Variables - Set 1

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A transport company provides bus, taxi and van service to send students to school and adults to work.

If a person takes the bus, another person takes taxi and a third person takes the van, then the total fare collected by the transport company is \(\text{RM}26\).

If a person takes the bus, \(2\) people take the taxi and \(3\) other people take the van, then the total fare collected is \(\text{RM}50\).

Finally, if \(2\) people take the bus, a person takes the taxi and another person takes the van, the the total fare is \(\text{RM}33\).

Find the bus, taxi and van fares for one person.

(A)		\(\begin{aligned} &\text{Bus fare}&\hspace{-0.5cm}=&\,\text{RM}8\\\\ &\text{Taxi fare}&\hspace{-0.5cm}=\,&\text{RM}15\\\\ &\text{Van fare}&\hspace{-0.5cm}=&\,\text{RM}6 \end{aligned}\)

(B)		\(\begin{aligned} &\text{Bus fare}&\hspace{-0.5cm}=&\,\text{RM}7\\\\ &\text{Taxi fare}&\hspace{-0.5cm}=\,&\text{RM}14\\\\ &\text{Van fare}&\hspace{-0.5cm}=&\,\text{RM}5 \end{aligned}\)

(C)		\(\begin{aligned} &\text{Bus fare}&\hspace{-0.5cm}=&\,\text{RM}8.\\\\ &\text{Taxi fare}&\hspace{-0.5cm}=\,&\text{RM}14.\\\\ &\text{Van fare}&\hspace{-0.5cm}=&\,\text{RM}5. \end{aligned}\)

(D)		\(\begin{aligned} &\text{Bus fare}&\hspace{-0.5cm}=&\,\text{RM}8.\\\\ &\text{Taxi fare}&\hspace{-0.5cm}=\,&\text{RM}14.\\\\ &\text{Van fare}&\hspace{-0.5cm}=&\,\text{RM}6. \end{aligned}\)

Difficulty Level: Medium

3.1 Systems of Linear Equations in Three Variables - Set 1

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Customer A buys \(3\) loaves of bread, \(5\) packets of biscuits and \(2\) boxes of milk with \(\text{RM}40\).

Customer B buys \(2\) loaves of bread, \(8\) packets of biscuits and \(3\) boxes of milk with \(\text{RM}55\).

Customer C buys \(4\) loaves of bread, \(7\) packets of biscuits and \(5\) boxes of milk with \(\text{RM}69\).

Find the price of a loaf of bread, a packet of biscuits and a box of milk.

(A)		\(\begin{aligned} &\text{Bread}&\hspace{-0.5cm}=&\,\text{RM}2.77\\\\ &\text{Biscuits}&\hspace{-0.5cm}=\,&\text{RM}3.87\\\\ &\text{Milk}&\hspace{-0.5cm}=&\,\text{RM}6.16 \end{aligned}\)

(B)		\(\begin{aligned} &\text{Bread}&\hspace{-0.5cm}=&\,\text{RM}3.77\\\\ &\text{Biscuits}&\hspace{-0.5cm}=\,&\text{RM}4.87\\\\ &\text{Milk}&\hspace{-0.5cm}=&\,\text{RM}7.16 \end{aligned}\)

(C)		\(\begin{aligned} &\text{Bread}&\hspace{-0.5cm}=&\,\text{RM}4.77\\\\ &\text{Biscuits}&\hspace{-0.5cm}=\,&\text{RM}5.87\\\\ &\text{Milk}&\hspace{-0.5cm}=&\,\text{RM}8.16 \end{aligned}\)

(D)		\(\begin{aligned} &\text{Bread}&\hspace{-0.5cm}=&\,\text{RM}5.77\\\\ &\text{Biscuits}&\hspace{-0.5cm}=\,&\text{RM}6.87\\\\ &\text{Milk}&\hspace{-0.5cm}=&\,\text{RM}9.16 \end{aligned}\)

Difficulty Level: Medium