Chapter 3.1, Problem 22PPSE

Solution

To Determine: Whether each system as independent, dependent, or inconsistent without graphing.

The system of equation is inconsistent.

Given information:

  $\left\{\begin{array}{l}2y=5x+6\hfill \\ -10x+4y=8\hfill \end{array}\right.$

Explanation:

If slope $m_1=m_2$ and $y-$ intercept $c_1=c_2$ , then the system of equations is consistent, dependent and it has infinitely many solutions.

If slope $m_1=m_2$ and $y-$ intercept $c_1\ne c_2$ , then the system of equations is inconsistent, it has no solutions.

If slope $m_1\ne m_2$ then the system of equations is consistent, independent and it has one solution.

Write the problem as a mathematical expression.

  $\begin{array}{c}2y=5x+6\to \left(1\right)\\ -10x+4y=8\to \left(2\right)\end{array}$

The equation $\left(1\right)$ is already in slope-intercept form $y=mx+b$ . So, divide the equation by $2$ :

  $y=\frac{5}{2}x+3\to \left(3\right)$

Now, add $10x$ to both sides of the equation $\left(2\right)$ :

  $\begin{array}{c}-10x+4y+10x=8+10x\\ 4y=8+10x\end{array}$

Rewrite the equation in slope-intercept form $y=mx+b$ :

  $4y=10x+8$

Divide the above equation by $4$ :

  $y=\frac{5}{2}x+2\to \left(4\right)$

Here, from the equations $\left(3\right)\&\left(4\right)$ slope $m_1=m_2$ and $y-$ intercept $c_1\ne c_2$ . Both equations have equal slopes, but their $y-$ intersections are different. In other words, the two lines are parallel. It indicates that the system is inconsistent and no solution.