Question

A) Graph the system (Let M be represented by the horizontal axis and n be represented by the vertical axis)B) Find the solution to the system

Answer 1

Let's work on the following equation

$n=\frac{2}{3}m+5$

when m = 0,

$\begin{gathered} n=\frac{2}{3}\cdot0+5 \\ n=0+5 \\ n=5 \end{gathered}$

... n = 5

therefore, one of the points of this line is (0, 5)

let's find a second point

when n = 0, then

$\begin{gathered} 0=\frac{2}{3}\cdot m+5 \\ \frac{2}{3}m+5=0 \\ \frac{2}{3}m+5-5=0-5 \\ \frac{2}{3}m=-5 \\ 3\cdot\frac{2}{3}m=3\mleft(-5\mright) \\ 2m=-15 \\ \frac{2m}{2}=\frac{-15}{2} \\ m=-\frac{15}{2} \\ m=-7.5 \end{gathered}$

... m = - 7.5

therefore, the second point for this line is (-7.5 , 0)

using this two points we can graph the system

so, the graph for the equation n = 2/3 m + 5 will be:

The second equation is:

$6m-9n=-45$

Again, let's find two points tha belong to the line

when m = 0, then

$\begin{gathered} 6\cdot0-9n=45 \\ 0-9n=-45 \\ -9n=-45 \\ n=\frac{45}{9} \\ n=5 \end{gathered}$

... n = 5

So, our first point for this equation, is (0. 5)

for our second point, when n = 0, then

$\begin{gathered} 6m-9\cdot0=-45 \\ 6m-0=-45 \\ 6m=-45 \\ m=\frac{-45}{6} \\ m=-7.5 \end{gathered}$

... m = -7.5

Therefore, the second point is (-7.5 , 0)

This equation can be represented by the following graph:

Finally, we need to solve the system of equations

$\begin{gathered} n=\frac{2}{3}m+5 \\ 6m-9n=-45 \end{gathered}$

substite n from eq1 to eq2

$6m-9\mleft(\frac{2}{3}m+5\mright)=-45$

simplify

$\begin{gathered} 6m-\frac{9\cdot2}{3}m-(9\cdot5)=-45 \\ 6m-6m-45=-45 \\ 0-45=-45 \\ -45=-45 \end{gathered}$

Since we obtained -45 = -45, the system of equations has infinite solutions