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.

$\left\{\begin{array}{l}y=2x-5\\ x=3y+4\end{array}\right.$

Short answer

The solution of the system is $\left(\frac{11}{5},-\frac{3}{5}\right)$.

Step-by-step solution

Step 1. Given Information 

We are given a system of equations $\left\{\begin{array}{l}y=2x-5\\ x=3y+4\end{array}\right.$.

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

Step 2. Substitute the value of y in second equation

Using the first equation $y=2x-5$, substitute $2x-5$for $y$in the second equation role="math" localid="1646968782205" $x=3y+4$.

$$\begin{gathered} x=3(2x-5)+4 \\ x=6x-15+4 \\ x=6x-11 \end{gathered}$$

Now solve the equation for $x$

$\begin{array}{rcl}x-6x& =& 6x-11-6x\\ -5x& =& -11\\ x& =& \frac{11}{5}\end{array}$

Step 3. Solve for y

Substitute $\frac{11}{5}$for $x$in the first equation and solve for $y$.

$$\begin{gathered} y=2x-5 \\ y=2·\frac{11}{5}-5 \\ y=\frac{22-25}{5} \\ y=-\frac{3}{5} \end{gathered}$$

So the solution is given by the ordered pair $\left(\frac{11}{5},-\frac{3}{5}\right)$.

Step 4. Check the solution  

Substitute $\frac{11}{5}$for $x$and $-\frac{3}{5}$for $y$in both the equations.

$$\begin{gathered} y=2x-5 \\ -\frac{3}{5}=2·\frac{11}{5}-5 \\ -\frac{3}{5}=\frac{22-25}{5} \\ -\frac{3}{5}=-\frac{3}{5} \\ x=3y+4 \\ \frac{11}{5}=3\left(-\frac{3}{5}\right)+4 \\ \frac{11}{5}=\frac{-9+20}{5} \\ \frac{11}{5}=\frac{11}{5} \end{gathered}$$

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