Q 335 - Intermediate Algebra

Question

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

$$\begin{gathered} 3x+2y=6 \\ y=-3x+4 \end{gathered}$$

Step-by-Step Solution

Answer

The system of equations with intersecting graphs has only one solution and is consistent and independent. 

Step 1. Given information

The given linear equations are 

$$\begin{gathered} 3x+2y=6 \\ y=-3x+4 \end{gathered}$$

Step 2. Finding the slopes of equations

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.

$$\begin{gathered} 3x + 2y = 6 \\ 2y = - 3x + 6 \\ y = - \frac{3}{2} + 3 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m = - \frac{3}{2} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} \mathrm{b} = 3 \\ \mathrm{Again} \mathrm{Solve} \mathrm{the} \mathrm{second} \mathrm{equation} \mathrm{for} \mathrm{y}. \mathrm{and} \mathrm{find} \mathrm{the} \mathrm{slope} \mathrm{and} \mathrm{y}-\mathrm{intercept}. \mathrm{y} = - 3x + 4 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} \mathrm{m}= -3 \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = 4 \end{gathered}$$

Step 3. Conclusion

The slopes are different here. As a result, the lines will intersect.

As a result, a system of equations with intersecting graphs has only one solution and is consistent and independent.