Why do equations with two distinct variables with 2 distinct

[navneet9431]

My question is about basic algebra. I am thinking about the "why" here and I'm looking for an intuitive answer.If you have the following equations:
=>S+U=90
=>S+U=90
and
=>40S+25U=2625
=>40S+25U=2625
you can then rewrite S=90−U and then substitute.

Now you have a single equation with one variable:
=>40(90−U)+25=2625
=>3600−40U+25U=2625
=>−15U=−975
Hence, U=65What's going on here? Ultimately, why does this always solve out? I realize single equations with one variable solve (there's got to be some number that satisfies this equation), but why? What's going on? I guess by solving the equation, we're bypassing this repetitive process of trial and error of plugging in numbers and seeing if it equals 2625? Is that what "solving the equation" really means?

Note: I am a High School student and English is my second language.Thanks!

 

[fresh_42]

My question is about basic algebra. I am thinking about the "why" here and I'm looking for an intuitive answer.If you have the following equations:
=>S+U=90
=>S+U=90
and
=>40S+25U=2625
=>40S+25U=2625
you can then rewrite S=90−U and then substitute.

Now you have a single equation with one variable:
=>40(90−U)+25=2625
=>3600−40U+25U=2625
=>−15U=−975
Hence, U=65What's going on here? Ultimately, why does this always solve out? I realize single equations with one variable solve (there's got to be some number that satisfies this equation), but why? What's going on? I guess by solving the equation, we're bypassing this repetitive process of trial and error of plugging in numbers and seeing if it equals 2625? Is that what "solving the equation" really means?

Note: I am a High School student and English is my second language.Thanks!

It does not always work. E.g. $2x+y=0$ and $4x+2y=0$ won't work. Geometrically you consider two straights here. They can cross each other as in your example, can be identical as in mine, or parallel. In the first case we find the point which is on both straights. It is like walking along one straight (equation 1) and then look where it crosses the other (equation 2, without leaving the first). In the second case (my example), you get back only one equation, that is one single, entire straight. And in the last case they don't have points in common. Parallelity results in a contradiction, e.g. $2x+y=2$ and $4x+2y=5$. No point $(x,y)$ satisfies both equations, i.e. lies on both straights if they are parallel.