Question

Fabiano and Sonali's teacher gave them a system of linear equations to solve. They each took a few steps that led to the systems shown in the table below.

Answer 1

Question: Which system has the same solution as the teacher's system?

Systems of equations that have the same solution are called equivalent systems.We are given with a system of two equations, we can make an equivalent system by substituting one equation by the sum of the two equations.

Given System:

                                 8x-16y=14→eq.1
                                 −x+5y=−3→eq.2
Teacher's Solution:
From eq.2
x = 5y + 3
Substituting the value of x in eq.1
We get,
 8(5y + 3) - 16y = 14
40y + 24 - 16y = 14
24y + 24 = 14
24y = 14-24
24y = -10
y = -10/24
y = -5/12

Substituting y-value in eq.2,
 −x+5(-5/12) = −3
-x -25/12 = -3
-x = -3 + 21/12
-x = 15/12
x = -5/4

Fabiano:
                                 3x−15y=9
                                 8x−16y=−7

Rearranging both equations:
y = 1/5x - 3/5
y = 1/2x+7/16

Placing both equations equal to each other,
1/5x - 3/5= 1/2x+7/16
1/5x -1/2x = 7/16 +3/5
-3/5x = 83/80
x = -83×5/80×3
x = -415/240
x = -83/48
This doesnt matches the teachers solution so we will move to sonali.

Sonali
                                 −x+5y=−3
                                 −4x+8y=−7
Rearranging both equations:
y = 1/5x - 3/5
y = 1/2x -7/8

Placing both equations equal to each other,
1/5x - 3/5 = 1/2x -7/8
1/5x -1/2x = -7/8 +3/5
-3/10x = -11/40
x = 110/120
x = 11/12

This doesn't match the teacher's solution either, so we will conclude that none of the solutions to the system of equation has the same answer as the teacher.

Answer 2

Fabiano and Sonali both have different solution with teacher’s solution.

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 Fabiano and Sonali.

The system of linear equation has given by the teacher is written below.

$\begin{aligned}8x+16y=14\\-x+5y=3\end{aligned}$

The system of linear equation has found by the Fabiano is written below.

$\begin{aligned}3x-15y=9\\8x-16y=-7\end{aligned}$

The system of linear equation has found by the Sonali is written below.

$\begin{aligned}-x+5y=-3\\-4x+8y=-7\end{aligned}$  

Step by step explanation:

Step 1:

Teacher’s solution:

First solve the equation has given by the teacher.

$\begin{aligned}8x-16y=14\\-x+5y=-3\end{aligned}$  

Now multiply the equation $-x+5y=-3$ with 8.

$- 8x + 40y = - 24$  

Now use elimination method to solve the system of equation.

$\begin{aligned}{\text{}}8x - 16y &= 14 \hfill \\\underline { - 8x + 40y &= - 24}\hfill \\{\text{24}}y &=  - 10 \hfill \\ \end{aligned}$  

Therefore, the value of $y$ can be calculated as,

$\begin{aligned}24y&=-10\\y&=-\dfrac{-10}{24}\\y&=-\dfrac{5}{12}\end{aligned}$  

Now substitute the value of $y=-\dfrac{5}{12}$ in to equation $-x+5y=-3$ to obtain the value of $x.$    

$\begin{aligned} - x + 5\left({ - \dfrac{5}{{12}}} \right) &= - 3\\-x-\dfrac{25}{12}&=-3\\x&=3-\dfrac{25}{12}\\x&=-\dfrac{5}{4}\end{aligned}$  

Therefore, the solution of teacher’s system of the equation is $\left(-\dfrac{5}{4},-\dfrac{5}{12}\right)$.

Step 2:

Fabiano’s solution

Now solve the equation has found by Fabiano.

$\begin{aligned}3x-5y&=9\\8x-16y&=-7\end{aligned}$  

Divide the first equation of the above system of linear equation by 3.

$\begin{aligned}\dfrac{3}{3}x-\dfrac{15}{3}y&=\dfrac{9}{3}\\x-5y&=6\end{aligned}$  

Divide the first equation of the above system of linear equation by $-8$.

$\begin{aligned}\dfrac{8}{-8}x-\dfrac{16}{-8}y&=\dfrac{-7}{-8}\\-x+2y&=\dfrac{7}{8}\end{aligned}$  

Now use elimination method to solve the system of equation.

$\begin{aligned}{\text{}}x - 5y &= 6 \hfill\\\underline { - x + 2y &= \frac{7}{8}}\hfill \\{\text{}} - 3y &= \dfrac{{41}}{8} \hfill \\\end{aligned}$  

The value of $y$ can be calculated as,

$\begin{aligned}-3y&=\frac{41}{8}\\y&=-\frac{41}{8}\times\frac{1}{3}\\y&=-\frac{41}{24}\end{aligned}$  

Therefore, the value of $y=-\dfrac{41}{24}$.

It can be seen that the solution of $y$ does not match with teacher’s solution therefore, there is no need to solve the further solution.

Step 3:

Selina’s solution:

Now solve the equation has found by Sonali.

$\begin{aligned}-x+5y&=-3\\-4x+8y&=-7\end{aligned}$  

The second equation of the above system can be written as,

$x-2y=\dfrac{7}{4}$  

Now use elimination method to solve the system of equation.

$\begin{aligned}- x + 5y &= - 3 \hfill \\\underline {{\text{}}x - 2y&= \frac{7}{4}}\hfill \\{\text{3}}y &=- \frac{5}{4} \hfill \\\end{aligned}$  

The value of $y$ can be calculated as,

$\begin{aligned}3y&=-\dfrac{5}{4}\\y&=-\dfrac{5}{4}\times\dfrac{1}{3}\\y&=-\dfrac{5}{12}\end{aligned}$  

Therefore, the value of $y=-\dfrac{5}{12}$.

Now substitute the value of $y=-\dfrac{5}{12}$ in to equation $x-2y=\dfrac{7}{4}$ to obtain the value of $x.$  

$\begin{aligned}x-2\left(-\dfrac{5}{12}\right)&=\dfrac{7}{4}\\x+2\left(\dfrac{5}{12}\right)&=\dfrac{7}{4}\\\x&=\dfrac{7}{4}-\dfrac{5}{6}\\x&=\frac{42-20}{24}\end{aligned}$  

Further simplify the above equation.

$\begin{aligned}x=\dfrac{22}{24}\\x=\dfrac{11}{12}\end{aligned}$  

The solution of Sonali’s system of the equation is $\left(\dfrac{11}{12,}-\dfrac{5}{12}\right)$.

Therefore, it can be seen that the solution of Sonali does not match with teacher’s solution.

Thus, both the students have different solution with teacher’s solution.

Learn more:  

1. Learn more about the solution of the linear equation https://brainly.com/question/11744034
2. Learn more about the problem of the linear equation https://brainly.com/question/2550559
3. Learn more about equations of the area and perimeter of the rectangle https://brainly.com/question/1511313

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Linear system of equations.

Keywords: Sonali, Fabiano, teacher, linear equation, system, elimination method, substitution, solution, multiply, divide, numbers, option.