Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.

9. $\begin{array}{c}x-2y=8\\ \frac{1}{2}x-y=4\end{array}$

Short answer

The system of equations isconsistent and dependent.

Step-by-step solution

Step-1 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-2 –Write the equations in slope-intercept form

Consider the first equation $x-2y=8$.

Subtract both sides by x.

$\begin{array}{c}x-2y-x=8-x\\ -2y=8-x\end{array}$

Divide both sides by $-2$.

$\begin{array}{c}y=\frac{8}{-2}-\frac{x}{-2}\\ y=-4+\frac{1}{2}x\\ y=\frac{1}{2}x-4\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{1}{2}$ and intercept of y-axis c is $-4$.

Now, consider the second equation $\frac{1}{2}x-y=4$.

Subtract both sides by $\frac{1}{2}x$.

$\begin{array}{c}\frac{1}{2}x-y-\frac{1}{2}x=4-\frac{1}{2}x\\ -y=4-\frac{1}{2}x\end{array}$

Divide both sides by $-1$.

$y=\frac{1}{2}x-4$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{1}{2}$ and intercept of y-axis c is $-4$.

Step-3 – Apply the concept of consistent and inconsistent system of equations

A system of equations is said to be consistent if there exist a solution to the system.

When two lines intersect each other at one point then there is only one solution to system of equations then the system is said to be consistent and independent.

When two lines are equivalent and their graph overlap each other then there are infinite solutions to system of equations then the system is said to be consistent and dependent.

A system of equations is said to be inconsistent if there no solution to the system that is two lines are parallel to each other.

Step-4 – Plot the lines and interpret

The blue line denotes the equation $x-2y=8$ and the equation $\frac{1}{2}x-y=4$.

The equations are equivalent and the same line denote the graph of each equation.

There are infinite solution to system of equations as all ordered pairs that lie on the line are the solution to system of equations.

Hence, the system of equations is consistent and dependent.