Step-by-step explanation:
Question-1:
by order pair we obtain:
[tex] \displaystyle \begin{cases} \displaystyle 3p = 2p - 1 \dots \dots i\\2q - p = 1 \dots \dots ii\end{cases}[/tex]
cancel 2p from the i equation to get a certain value of p:
[tex] \displaystyle \begin{cases} \displaystyle p = - 1 \\2q - p = 1 \end{cases}[/tex]
now substitute the value of p to the second equation:
[tex] \displaystyle \begin{cases} \displaystyle p = - 1 \\2q - ( - 1) = 1 \end{cases}[/tex]
simplify parentheses:
[tex] \displaystyle \begin{cases} \displaystyle p = - 1 \\2q + 1= 1 \end{cases}[/tex]
cancel 1 from both sides:
[tex] \displaystyle \begin{cases} \displaystyle p = - 1 \\2q = 0\end{cases}[/tex]
divide both sides by 2:
[tex] \displaystyle \begin{cases} \displaystyle p = - 1 \\q = 0\end{cases}[/tex]
question-2:
by order pair we obtain:
[tex] \displaystyle \begin{cases} \displaystyle 2x - y= 3 \dots \dots i\\3y= x + y \dots \dots ii\end{cases}[/tex]
cancel out y from the second equation:
[tex] \displaystyle \begin{cases} \displaystyle 2x - y= 3 \dots \dots i\\ x = 2y \dots \dots ii\end{cases}[/tex]
substitute the value of x to the first equation:
[tex] \displaystyle \begin{cases} \displaystyle 2.2y-y= 3 \\ x = 2y \end{cases}[/tex]
simplify:
[tex] \displaystyle \begin{cases} \displaystyle 3y= 3 \\ x = 2y \end{cases}[/tex]
divide both sides by 3:
[tex] \displaystyle \begin{cases} \displaystyle y= 1 \\ x = 2y \end{cases}[/tex]
substitute the value of y to the second equation which yields:
[tex] \displaystyle \begin{cases} \displaystyle y= 1 \\ x = 2 \end{cases}[/tex]
Question-3:
by order pair we obtain;
[tex] \displaystyle \begin{cases} \displaystyle 2p + q = 2 \dots \dots i\\3q + 2p = 3 \dots \dots ii\end{cases}[/tex]
rearrange:
[tex] \displaystyle \begin{cases} \displaystyle 2p + q = 2 \\2p + 3q= 3 \end{cases}[/tex]
subtract and simplify
[tex] \displaystyle \begin{array}{ccc} \displaystyle 2p + q = 2 \\2p + 3q= 3 \\ \hline - 2q = - 1 \\ q = \dfrac{1}{2} \end{array}[/tex]
substitute the value of q to the first equation:
[tex] \displaystyle 2.p+ \frac{1}{2} = 2[/tex]
make q the subject of the equation:
[tex] \displaystyle p = \frac{3}{4} [/tex]
hence,
[tex] \displaystyle q = \frac{1}{2} \\ p = \frac{3}{4} [/tex]
