Chapter 12, Problem 1RE

To determine

The solution set of the system of linear equations $\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}$.

Answer to Problem 1RE

The solution set of linear equations $\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}$ is $\underset{\_}{\left\{\left(2,-1\right)\right\}}$.

Explanation of Solution

Given:

The system of linear equations is $\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}$.

Procedure used:

“Solving system of linear equations by elimination:

(1) If the equations are not in the standard form, then they are converted accordingly into a standard form.

(2) The variable to be eliminated is recognized from the given equations in the system.

(3) After obtaining the LCM (lowest common multiple) of coefficients of the variable to be eliminated, the equations are multiplied with appropriate multipliers so that the coefficients in each equation become additive inverses and adding the new equations will result in a fresh equation with the decided variable to been eliminated.

(4) The equation in step (3) is solved obtaining value of remaining variable.

(5) The value of variable in step (4) is substituted to the equation in any of the given system equations.

(6) Solving the equation obtained in step (5) gives the value of variable eliminated in step (3).

(7) The correctness of solution is asserted by substituting back the values in given equations of the system.

Calculation:

The given equations are, as follows:

$\begin{array}{l}2x-y=5\mathrm{......}\left(1\right)\\ 5x+2y=8\mathrm{......}\left(2\right)\end{array}$

Step 1:

Since the given equations are already in standard form, nothing has to be done.

$\begin{array}{c}\left(2\right)x-\left(1\right)y=5\\ \left(5\right)x+\left(2\right)y=8\end{array}$

Step 2:

The coefficient of y in equation (1) is $-1$ and in equation (2) is $2$. So, $y$ is a better choice of variable to be eliminated.

Step 3:

Multiply equation (1) with number $2$ to eliminate the variable $y$, as follows:

$\begin{array}{l}2\left(2x-y\right)=2\cdot 5\\ 4x-2y=10\end{array}$

Now, add resultant equation of equation (1) and equation (2) of given system of linear equations:

$\begin{array}{l}4x-2y=10\\ \underset{\_}{5x+2y=8}\\ 9x=18\end{array}$

The above addition gives the equation $9x=18$.

Step 4:

Solve the equation obtained in step 3 to obtain the value of $x$.

$\begin{array}{c}x=\frac{18}{9}\\ =2\end{array}$

Thus, the value of x is $2$.

Step-5:

Now, substitute $2$ for $x$ in equation (1) to obtain the corresponding value of $y$.

$\begin{array}{l}2\left(2\right)-y=5\\ 4-y=5\\ y=4-5\\ y=-1\end{array}$

Step-6:

The equation obtained in Step (5) is solved:

$y=-1$

Thus, the value of y is $-1$.

Step-7:

In order to check whether the solution is correct or not, substitute $x=2$ and $y=-1$ in given system of equation.

Substitute $2$ for $x$, $-1$ for $y$ in the equation (1) to check whether the solution $\left(2,-1\right)$ satisfies the equation (1) or not.

$\begin{array}{l}2\left(2\right)-\left(-1\right)\stackrel{?}{=}5\\ 4+1\stackrel{?}{=}5\\ 5=5\left(\text{True}\right)\end{array}$

Therefore, the point $\left(2,-1\right)$ satisfies equation (1).

Substitute $2$ for $x$, $-1$ for $y$ in the equation (2) to check whether the solution $\left(2,-1\right)$ satisfies the equation (2) or not.

$\begin{array}{l}5\left(2\right)+2\left(-1\right)\stackrel{?}{=}8\\ 10-2\stackrel{?}{=}8\\ 8=8\left(\text{True}\right)\end{array}$

Therefore, the point $\left(2,-1\right)$ satisfies equation (2).

Hence, it is asserted that the value of x is $2$ and the value of y is $-1$.

Thus, the solution set for the system of linear equations $\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}$ is $\underset{\_}{\left\{\left(2,-1\right)\right\}}$.