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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:

$$S + U = 90$$ and $$40S + 25U = 2625$$

you can then rewrite $S = 90 - U$ and then substitute.

Now you have a single equation with one variable:

$$\begin{align*} 40(90 - U) + 25& = 2625 \\ 3600 - 40U + 25U& = 2625 \\ -15U& = -975 \\ U& = 65 \end{align*}$$

What's going on here? Ultimately, why does this always solve out? I realize single equations with one variable solve (there's gotta be some number that satisfies this equation), but why? What's going on? I guess by solving the equation, we're bypassing this iterative process of trial and error of plugging in numbers and seeing if it equals 2625? Is that what "solving the equation" really means?

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  • $\begingroup$ It's possible to have a system of two linear equations with two unknowns that has no solutions. But if you write your system as $A x = b$, and recognize that we are looking for a linear combination of the columns of $A$ that is equal to $b$, then you can see that typically the columns of $A$ will be linearly independent, hence $b$ is in the span of the columns of $A$ and $Ax = b$ has a solution. $\endgroup$ – littleO Dec 16 '13 at 3:24
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It can help to picture the equations as functions on a graph, and looking for where they intersect, because those intersections are the solutions to the system of equations.

One variable is a degenerate case, so let's skip that.

With two variables, you have a 2D space and each equation defines a line. Those two lines will intersect in one spot*, which is your solution.

With three variables, you're looking at a 3D space and each equation defines a surface. Two surfaces intersect in a line*, which isn't a solution. You need a third surface (a third equation) that will intersect that line in one spot*, which is your solution.

This continues into higher dimensional spaces with more variables.

* if any of these things are parallel then you will not get an intersection, which means your system of equations does not have a solution. Or the things could be coincident, in which case you don't have enough information to find the solution.

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A single equation in one variable picks out a value in $\Bbb R$ An equation in two variables picks out a line in $\Bbb R^2$. Two equations pick out two lines, which generally intersect in a point. Basically each equation eliminates one degree of freedom, so you need as many equations as unknowns.

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It doesn't "always solve out" - This only works so nicely because the equations are linear, and moreover not dependant on one another (as we'll explain). Imagine an equation like: $$y=5x +2$$

This describes a relationship between $y$ and $x$ - one that cab be drawn on a graph as a straight line. For a non parallel to the axis straight line, each $y$ can correspond to only one $x$ and vice versa.

Then introduce a second linear equation, which again describes a straight line. A solution is exactly where these lines meet, but of course there are some lines which will not meet - and thus there will be no solution.

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