Question

Solve the following system of equations by graphing and state whether the system is dependent, independent, or consistent. 1/2x + 3/4y = 12x - 3y = 4

Answer 1

We have to solve the following system of equations:

`\begin{gathered} (1)/(2)x+(3)/(4)y=1 \\ 2x-3y=4 \end{gathered}`

We have to graph the equations and, as they are written in standard form, we are going to calculate the intercepts for both.

We will write the equations in slope-intercept form.

For the first equation we have:

`\begin{gathered} (1)/(2)x+(3)/(4)y=1 \\ (3)/(4)y=-(1)/(2)x+1 \\ y=(4)/(3)(-(1)/(2)x+1) \\ y=-(2)/(3)x+(4)/(3) \end{gathered}`

For the second equation we have:

`\begin{gathered} 2x-3y=4 \\ 2x-4=3y \\ y=(1)/(3)(2x-4) \\ y=(2)/(3)x-(4)/(3) \end{gathered}`

Both slopes are different, what means that the lines are not parallel and will intersect, so we already know that the system is independent.

Using the slopes and the y-intercepts, we can graph the equations as:

The solution to the system is the intersection point which is (2,0).

Answer:

The system is independent and its solution is (x,y) = (2,0)