We begin with 2m+3n=4 and -m+2n=5. I prefer to solve these kinds of problems by elimination. We could also solve this by substitution or by graphing.
The way we set up elimination is like so:
color(white)(.)2m+3n=4
-m+2n=5
Now, the goal of elimination is to get rid of a variable. Right now, nothing would subtract or add to zero, so let's tinker with the problem. I'm going to multiply -m+2n=5 by 2 so that I can add away the ms. I'll show you what I mean:
color(white)(..............)2m+3n=4
+
color(white)(.....)2*(-m+2n=5)
Which becomes
color(white)(.....)2m+3n=4
+
color(white)(.)-2m+4n=10
-------
color(white)(........)0+7n=14
Divide by 7 on both sides, and we have n=2
To find m, we just plug in n=2 to one of the equations. I'm just going to use -m+2n=5. We have -m+2(2)=5, or -m+4=5, which we can simplify to -m=1. If we divide by -1 on both sides, we have m=-1.
To double check our work, let's plug our values for m and n into the other equation, 2m+3n=4. Once we plug in m and n for -1 and 2, respectively, we have 2(-1)+3(2)=4. This becomes -2+6, which equals 4. 4=4, and we got it right! Nice job!
