Chapter 6, Problem 11PT

To determine

To solve: The system of equations using elimination method.

Answer to Problem 11PT

The solution of given system of equations is $x=3$ and $y=-1$ .

Explanation of Solution

Given information:

The given system of equations is

  $3x+7y=2$

  $3x-4y=13$

Method used:

The steps of performing elimination method are:

Step 1: Write each given equation in standard form $Ax+By=C$ and $A^{\prime }x+B^{\prime }y=C^{\prime }$ .

Step 2: Equate coefficients of one variable equal by multiplying coefficients of variables in both equations. Let for variable “ x ”

  $A^{\prime }Ax+A^{\prime }By=A^{\prime }C$ , and $A{A}^{\prime }x+A{B}^{\prime }y=A{C}^{\prime }$ , then subtracting one equation from other variable “ x ” gets eliminated and remaining equation can be solved for “ y ”.

Calculations:

The given equations are

  $3x+7y=2$  ...... (1)

  $3x-4y=13$   ...... (2)

As the coefficients of variable “ x ” in both equations are identical and of same sign so subtracting equations (2) from equation (1)

  $\left(3x+7y\right)-\left(3x-4y\right)=2-13$

  $\Rightarrow \cancel{3x}-\cancel{3x}+7y-(-4y)=-11$

  $\Rightarrow 11y=-11$

  $\Rightarrow y=\frac{-11}{11}=-1$

Further, substituting value of y in equation (1)

  $3x+7(-1)=2$

  $\Rightarrow 3x-7=2$

  $\Rightarrow 3x=2+7$

  $\Rightarrow 3x=9$

  $\Rightarrow x=\frac{9}{3}=3$

Thus, $x=3$ , $y=-1$ is the solution of system of equations $3x+7y=2$ and $3x-4y=13$ .

Conclusion:

The solution of given system of equations is $x=3$ and $y=-1$ .