Chapter 3.2, Problem 13PPSE

Solution

To solve the system of equations $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ by the method of substitution and to check whether the answer is right.

The solution of the system of equations $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ is found and verified to be $p=\frac{3}{4}$ and $q=\frac{5}{2}$ .

Given the system of equations:

  $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ .

Concept Used:

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

Given a system of linear equations in two variables:

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

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

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

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

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

Calculation:

Solve the system of equations by the method of substitution:

Observe that the second equation $q=2p+1$ is already suitable for being substituted in the first one.

Substitute $q=2p+1$ in the first equation $4p+2q=8$ and solve for $p$ :

  $\begin{array}{c}4p+2\left(2p+1\right)=8\\ 4p+4p+2=8\\ 4p+4p=8-2\\ 8p=6\\ p=\frac{6}{8}\\ p=\frac{3}{4}\end{array}$

Thus, it is found that:

  $p=\frac{3}{4}$ .

Substitute $p=\frac{3}{4}$ in the first equation $4p+2q=8$ and solve for $q$ :

  $\begin{array}{c}4p+2q=8\\ 4\left(\frac{3}{4}\right)+2q=8\\ 3+2q=8\\ 2q=8-3\\ 2q=5\\ q=\frac{5}{2}\end{array}$ .

Thus, it is found that:

  $q=\frac{5}{2}$ .

Thus, the solution of the system of equations $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ :

  $p=\frac{3}{4}$ and $q=\frac{5}{2}$ .

Check whether the answer is correct or not:

Substitute $p=\frac{3}{4}$ and $q=\frac{5}{2}$ in the first equation $4p+2q=8$ and determine whether it gives a valid equation:

  $\begin{array}{c}4p+2q=8\\ 4\left(\frac{3}{4}\right)+2\left(\frac{5}{2}\right)=8\\ 3+5=8\\ 8=8\end{array}$ .

It is observed that $4p+2q=8$ gives a valid equation when it is substituted with $p=\frac{3}{4}$ and $q=\frac{5}{2}$ .

Thus, $p=\frac{3}{4}$ and $q=\frac{5}{2}$ satisfy the equation $4p+2q=8$ .

Substitute $p=\frac{3}{4}$ and $q=\frac{5}{2}$ in the second equation $q=2p+1$ to check whether it gives a valid equation:

  $\begin{array}{c}q=2p+1\\ \left(\frac{5}{2}\right)=2\left(\frac{3}{4}\right)+1\\ \frac{5}{2}=\frac{3}{2}+1\\ \frac{5}{2}=\frac{3}{2}+\frac{2}{2}\\ \frac{5}{2}=\frac{5}{2}\end{array}$ .

It is observed that $q=2p+1$ gives a valid equation when it is substituted with $p=\frac{3}{4}$ and $q=\frac{5}{2}$ .

Thus, $p=\frac{3}{4}$ and $q=\frac{5}{2}$ satisfy the equation $q=2p+1$ .

Hence, $p=\frac{3}{4}$ and $q=\frac{5}{2}$ satisfy both the equations $4p+2q=8$ and $q=2p+1$ .

That is, $p=\frac{3}{4}$ and $q=\frac{5}{2}$ is a solution of the system of equations $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ .

Hence, the answer so found is correct.

Conclusion:

The solution of the system of equations $\left\{\begin{array}{c}4p+2q=8\\ q=2p+1\end{array}\right.$ is found and verified to be $p=\frac{3}{4}$ and $q=\frac{5}{2}$ .