Chapter 6.3, Problem 14P

Error Analysis A student solved a system of linear equations using the elimination method as follows. Describe and correct the error made by the student.

$3x-5y=4$ $-2x+3y=2$

$6x-10y=8$ $\frac{-6x+3y=6}{-7y=14}$ $y=-2$

Multiply equation 1 by 2. Multiply equation 2 by 3. Add the equations. Divide by -7

To determine

To find: the error and correct it with describing it.

Answer to Problem 14P

  $x=-22,y=-14$

Mistake of student:- In Equation (2) student multiply by 3 but didn’t change $y$ ’s coefficient (from 3 $y$ to 9 $y$ )

Explanation of Solution

Given information:

Given system of equation is

  $3x-5y=4$

  $-2x+3y=2$

Calculation:- Here given system of equation is

  $3x-5y=4$ ---- (1)

  $-2x+3y=2$ ------ (2)

Now, to apply elimination method in system of linear equation, one variable (either $x$ or $y$ ) should have same coefficient in both the equations. Then eliminate that variable from both equations.

In given system of equation,

  $6x-10y=8$ Multiplying by 2 in equation (1)

  $-6x+9y=6$ Multiplying by 3 in equation (2)

Now, we can see $x$ has same coefficient in both the equation. To eliminate $x$ from both the equations, add the equations and we get-

  $\begin{array}{l}6x-10y=8\\ -6x+9y=6\text{ add the equation}\\ \overline{\text{ -y = 14 }}\\ \text{ y = -14}\end{array}$

To find the value of $x$ , put $y$ = -14 in either equation (1) or (2).

  $\begin{array}{l}3x-5y=4\\ 3x-5(-14)=4\\ 3x+70=4\\ 3x=4-70\\ 3x=-66\\ x=-22\end{array}$

Mistake of student:- In Equation (2) student multiply by 3 but didn’t change $y$ ’s coefficient (from 3 $y$ to 9 $y$ )

Verification:-

To verify values of x and y put these in given system, if both the side of equations are same then values are correct.

  $\begin{array}{l}3x-5y=4\\ 3(-22)-5(-14)=4\\ -66+70=4\\ 4=4\\ \end{array}$  $\begin{array}{l}-2x+3y=2\\ -2(-22)+3(-14)=2\\ 44-42=2\\ 2=2\end{array}$

Hence values of $x$ and $y$ are verified.