Q 334 - Intermediate Algebra

Question

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

$$\begin{gathered} \left\{\begin{array}{l} y= \frac{2}{5}x+2 \\ -2x+5y=10\\ \end{array}\right. \end{gathered}$$

Step-by-Step Solution

Answer

The  system of equations with parallel lines as graphs has no solution, is inconsistent, and independent.

Step 1. Given information

The given linear equations are

$$\begin{gathered} \left\{\begin{array}{l} y= \frac{2}{5}x+2 \\ -2x+5y=10\\ \end{array}\right. \end{gathered}$$

Step 2. Finding intersection points

First, solve both of these equations for y such that their slopes and y intercepts may be easily graphed.

And find the slope and y-intercept by solving the first equation for y.

$$\begin{gathered} y = \frac{2}{5} x + 2 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} \mathrm{m} =\frac{2}{5} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} \mathrm{b} = 2 \\ \mathrm{Again} \mathrm{Solve} \mathrm{the} \mathrm{second} \mathrm{equation} \mathrm{for} \mathrm{y}. \mathrm{and} \mathrm{find} \mathrm{the} \mathrm{slope} \mathrm{and} \mathrm{y}-\mathrm{intercept}. - 2x + 5y = 10 \\ 5y = 2x + 10 \\ y = \frac{2}{5}x + 2 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m =\frac{2}{5} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{i}s b = 2 \end{gathered}$$

Step 3. Conclusion

As a result, a system of equations with parallel lines as graphs has no solution, is inconsistent, and independent.