Q. 1 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent. 

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

Step-by-Step Solution

Answer

The solution of the system is $\left(2,-1\right)$.

Step 1. Given Information

We are given a system of equation $\left\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}\right.$.

We need to solve the system using the method of substitution or the method of elimination.

Step 2. Eliminate the variable y and solve for x

Multiply both sides of the first equation $2x-y=5$ by $2$

$\begin{array}{rcl}2(2x-y)& =& 2·5\\ 4x-2y& =& 10\end{array}$

Now add this equation with the first equation and solve for $x$.

$\begin{array}{rcl}5x+2y+(4x-2y)& =& 8+10\\ 5x+2y+4x-2y& =& 18\\ 9x& =& 18\\ x& =& \frac{18}{9}\\ x& =& 2\end{array}$

Step 3. Solve for y

Substitute $2$ for $x$ in the first equation and solve for $y$.

$\begin{array}{rcl}2x-y& =& 5\\ 2·2-y& =& 5\\ 4-y& =& 5\\ -y& =& 5-4\\ -y& =& 1\\ y& =& -1\end{array}$

So the solution of the system is the ordered pair $\left(2,-1\right)$.

Step 4. Check the solution

Substitute $2$ for $x$ and $-1$ for $y$ in both the equations.

$\begin{array}{rcl}2x-y& =& 5\\ 2·2-(-1)& =& 5\\ 4+1& =& 5\\ 5& =& 5\\ & & \\ 5x+2y& =& 8\\ 5·2+2(-1)& =& 8\\ 10-2& =& 8\\ 8& =& 8\end{array}$

So both the equations are satisfied. Thus the found solution is correct.