The correct option is [tex]\boxed{{\mathbf{Option D}}}[/tex].
Further explanation:
The system of the linear equation can be solved by the elimination method and substitution method.
Given:
Teacher gave a system of linear equation to her students Goku and Selina.
The system of linear equation has given by the teacher is written below.
[tex]\begin{aligned}- 2x + 5y &= 10 \hfill\\-3x + 9y&= 6 \hfill\\\end{aligned}[/tex]
The system of linear equation has found by the Goku is written below.
[tex]\begin{aligned}x - 3y &= - 2 \hfill\\- 2x + 5y &=- 7 \hfill\\\end{aligned}[/tex]
The system of linear equation has found by the Selina is written below.
[tex]\begin{aligned}- 5x + 14y &= 16 \hfill \\- 3x + 9y &= 12 \hfill \\\end{aligned}[/tex]
Step by step explanation:
Step 1:
First solve the equation has given by the teacher.
[tex]\begin{aligned}- 2x + 5y &= 10 \hfill \\- 3x + 9y &= 6 \hfill\\\end{aligned}[/tex]
The second equation of the above system can be written as,
[tex]- x + 3y = 2[/tex]
Now multiply the equation [tex]- x + 3y = 2[/tex] with 2.
[tex]- 2x + 6y = 4[/tex]
Now use elimination method to solve the system of equation.
[tex]\begin{aligned}- 2x + 6y &= 4\hfill\\\underline { - 2x + 5y &= 10}\hfill\\{\text{ }}y& = - 6 \hfill\\\end{aligned}[/tex]
Now substitute the value of [tex]y = - 6[/tex] in to equation [tex]- 2x + 5y = 10[/tex] to obtain the value of [tex]x[/tex]
[tex]\begin{aligned}- 2x + 5\left( { - 6} \right) &= 10\\- 2x - 30 &= 10\\- 2x &= 10 + 30\\x &= - 20 \\\end{aligned}[/tex]
Therefore, the solution of teacher’s system of the equation is [tex]\left( { - 20, - 6} \right)[/tex].
Step 2:
Now solve the equation has found by Goku.
[tex]\begin{aligned}x - 3y&= - 2 \hfill\\- 2x + 5y &= - 7 \hfill\\\end{aligned}[/tex]
Now multiply the equation [tex]x - 3y = - 2[/tex] with 2.
[tex]2x - 6y = - 4[/tex]
Now use elimination method to solve the system of equation.
[tex]\begin{aligned}{\text{ }}2x - 6y &= - 4 \hfill\\\underline { - 2x + 5y &= - 7} \hfill \\{\text{ }} - y &= - 11 \hfill \\\end{aligned}[/tex]
Therefore, the value of [tex]y = 11[/tex].
Now substitute the value of [tex]y = 11[/tex] in to equation [tex]x - 3y = - 2[/tex] to obtain the value of [tex]x[/tex].
[tex]\begin{aligned}x - 3\left( {11} \right) &= - 2\\x - 33 &= - 2\\x &= - 2 + 33\\x &= 31\\\end{aligned}[/tex]
Therefore, the solution of Goku’s system of the equation is [tex]\left( {31,11} \right)[/tex].
Step 3:
Now solve the equation has found by Selina.
[tex]\begin{aligned}- 5x + 14y&= 16 \hfill\\- 3x + 9y &= 12 \hfill\\\end{aligned}[/tex]
The second equation of the above system can be written as,
[tex]\begin{aligned}- 3x + 9y&= 12\\3\left({ - x + 3y} \right) &= 3\left( 4 \right)\\- x + 3y &= 4\\\end{aligned}[/tex]
Now multiply the equation [tex]- x + 3y = 4[/tex] with [tex]- 5[/tex].
[tex]\begin{aligned}- \left({ - 5} \right)x + \left({ - 5} \right)3y &= \left( { - 5} \right)4\\5x - 15y&= - 20\\\end{aligned}[/tex]
Now use elimination method to solve the system of equation.
[tex]\begin{aligned}- 5x + 14y &= 16 \hfill \\\underline {{\text{ }}5x - 15y &= - 20} \hfill\\ {\text{ }} - y &=- 4 \hfill\\\end{aligned}[/tex]
Therefore, the value of [tex]y = 4[/tex] .
Now substitute the value of [tex]y = 4[/tex] in to equation [tex]- 5x + 14y = 16[/tex] to obtain the value of
[tex]\begin{aligned}- 5x + 14\left( 4 \right) &= 16\\- 5x + 56 &= 16\\- 5x &= 16 - 56\\- 5x &= - 40\\x &= 8\\\end{aligned}[/tex]
Therefore, the solution of Selina’s system of the equation is [tex]\left( {16,4} \right)[/tex].
Thus, it can be seen that neither Goku nor Selina has the same solution as the teacher system.
Therefore, the correct option is [tex]\boxed{{\mathbf{Option D}}}[/tex].
Learn more:
Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Linear system of equations.
Keywords: Selina, Goku, teacher, linear equation, system, elimination method, substitution, solution, multiply, divide, numbers, option
