Chapter 8, Problem 1RE

To determine

To solve: The system of equation $\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}$.

Answer to Problem 1RE

The solution of the system of equation is $\underset{\_}{x=2\text{ and }y=-1}$.

Explanation of Solution

Procedure used:

Step1: Pick one of the equations and solve for one of the variables in terms of the remaining variables.

Step2: Substitute the result into the remaining equations.

Step3: If one equation in one variable results, solve this equation. Otherwise, repeat steps 1 and 2 until a single equation with one variable remains.

Step4: Find the values of the remaining variables by back substitutions.

Step5: Check the solution found.

Calculation:

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

Solve the equation by substitution method.

Solve $2x-y=5$ for y as follows.

$\begin{array}{l}2x-y=5\\ y=2x-5\end{array}$

Substitute $y=2x-5$ in the equation $5x+2y=8$.

$\begin{array}{l}5x+2y=8\\ 5x+2\left(2x-5\right)=8\\ 5x+4x-10=8\\ 9x-10=8\end{array}$

On further simplification,

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

Substitute $x=2$ in the equation $5x+2y=8$.

$\begin{array}{l}5x+2y=8\\ 5\left(2\right)+2y=8\\ 10+2y=8\\ 2y=8-10\end{array}$

On further simplification,

$\begin{array}{l}y=\frac{-2}{2}\\ y=-1\end{array}$

Check by substituting $x=2\text{ and }y=-1$ in the equation $2x-y=5$.

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

Check by substituting $x=2\text{ and }y=-1$ in the equation $2x-y=5$.

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

Thus, the solution of the system of equation is $\underset{\_}{x=2\text{ and }y=-1}$.