Question

Without graphing, determine the number of solutions and then classify the system of equations.

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

Short answer

The number of solution is one and the system of equations is intersect lines, consistent and independent.

Step-by-step solution

Step 1. Given information

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

Step 2. Solve the first equation for y and find the slope and y-intercept. 

$$\begin{gathered} 5x+3y=4 \\ 3y=-5x+4 \\ y=-\frac{5}{3}x+\frac{4}{3} \end{gathered}$$

Here the slope is role="math" $-\frac{5}{3}$

And the$y$-intercept is$\frac{4}{3}$

Step 3. Solve the second equation for y and find the slope and y-intercept. 

$$\begin{gathered} 2x-3y=5 \\ 3y=2x-5 \\ y=\frac{2}{3}x-\frac{5}{3} \end{gathered}$$

Step 4. Required answer

Here the slopes are different.

Therefore, the system of equation is intersect lines with one solution and is consistent and independent.