Chapter 0.1, Problem 36E

To determine

To find the pair $\left(x,y\right)$ that satisfies both the equations $2y-5x=0$ and $4x+y=26$ simultaneously.

Answer to Problem 36E

The pair $\left(x,y\right)$ that satisfies both the equations $2y-5x=0$ and $4x+y=26$ simultaneously:

  $\left(x,y\right)=\left(4,10\right)$ .

Explanation of Solution

Given the equations:

  $2y-5x=0$ and $4x+y=26$ .

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 $2y-5x=0$ and $4x+y=26$ simultaneously, it is enough the solve the system of equations $\left\{\begin{array}{c}2y-5x=0\\ 4x+y=26\end{array}\right.$ for $x$ and $y$ .

Solve the system of equations:

Consider the second equation $4x+y=26$ .

Isolate $y$ on the left side of the equation $4x+y=26$ :

  $\begin{array}{c}4x+y=26\\ y=26-4x\end{array}$ .

Substitute $y=26-4x$ in the first equation $2y-5x=0$ and solve for $x$ :

  $\begin{array}{c}2y-5x=0\\ 2\left(26-4x\right)-5x=0\\ 52-8x-5x=0\\ -8x-5x=-52\\ -13x=-52\\ x=\frac{-52}{-13}\\ x=4\end{array}$ .

Thus, it is found that:

  $x=4$ .

Substitute $x=4$ in the second equation $4x+y=26$ and solve for $y$ :

  $\begin{array}{c}4x+y=26\\ 4\left(4\right)+y=26\\ 16+y=26\\ y=26-16\\ y=10\end{array}$ .

Thus, it is found that:

  $y=10$ .

Thus, the solution of the system of equations $\left\{\begin{array}{c}2y-5x=0\\ 4x+y=26\end{array}\right.$ :

  $x=4$ and $y=10$ .

That is, the pair $\left(x,y\right)$ that satisfies both the equations $2y-5x=0$ and $4x+y=26$ simultaneously:

  $\left(4,10\right)$ .

Conclusion:

The pair $\left(x,y\right)$ that satisfies both the equations $2y-5x=0$ and $4x+y=26$ simultaneously:

  $\left(x,y\right)=\left(4,10\right)$ .