Chapter 0.1, Problem 34E

To determine

To find the pair $\left(x,y\right)$ that satisfies both the equations $2x+y=11$ and $6x-y=5$ simultaneously.

Answer to Problem 34E

The pair $\left(x,y\right)$ that satisfies both the equations $2x+y=11$ and $6x-y=5$ simultaneously:

  $\left(x,y\right)=\left(2,7\right)$ .

Explanation of Solution

Given the equations:

  $2x+y=11$ and $6x-y=5$ .

Concept Used:

Method of substitution in the case of linear equations involving two variables:

Given a system of linear equations in two variables:

  $\left\{\begin{array}{c}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}\right.$ .

The method of substitution employs solving one of the equations, say $a_{1}x+b_{1}y=c_1$ , to find an algebraic expression for one of its variables, say $x$ .

Now, this expression of $x$ is substituted in the other equation $a_{2}x+b_{2}y=c_2$ and then solved to find the value of $y$ .

The value of $y$ is now substituted in either $a_{1}x+b_{1}y=c_1$ or $a_{2}x+b_{2}y=c_2$ to find the value of $x$ .

Now, the value of $x$ and that of $y$ are recorded to be the solution of the system of equations.

Calculation:

To find the pair $\left(x,y\right)$ that satisfies both the equations $2x+y=11$ and $6x-y=5$ simultaneously, it is enough the solve the system of equations $\left\{\begin{array}{c}2x+y=11\\ 6x-y=5\end{array}\right.$ for $x$ and $y$ .

Solve the system of equations:

Consider the first equation $2x+y=11$ .

Isolate $y$ on the left side of the equation $2x+y=11$ :

  $\begin{array}{c}2x+y=11\\ y=11-2x\end{array}$ .

Substitute $y=11-2x$ in the second equation $6x-y=5$ and solve for $x$ :

  $\begin{array}{c}6x-y=5\\ 6x-\left(11-2x\right)=5\\ 6x-11+2x=5\\ 6x+2x=5+11\\ 8x=16\\ x=\frac{16}{8}\\ x=2\end{array}$ .

Thus, it is found that:

  $x=2$ .

Substitute $x=2$ in the second equation $6x-y=5$ and solve for $y$ :

  $\begin{array}{c}6x-y=5\\ 6\left(2\right)-y=5\\ 12-y=5\\ -y=5-12\\ -y=-7\\ y=\frac{-7}{-1}\\ y=7\end{array}$ .

Thus, it is found that:

  $y=7$ .

Thus, the solution of the system of equations $\left\{\begin{array}{c}2x+y=11\\ 6x-y=5\end{array}\right.$ :

  $x=2$ and $y=7$ .

That is, the pair $\left(x,y\right)$ that satisfies both the equations $2x+y=11$ and $6x-y=5$ simultaneously:

  $\left(2,7\right)$ .

Conclusion:

The pair $\left(x,y\right)$ that satisfies both the equations $2x+y=11$ and $6x-y=5$ simultaneously:

  $\left(x,y\right)=\left(2,7\right)$ .