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# Equations and Unknowns

Sometimes $$x$$ is unknown, and sometimes it's unknowable.

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This chapter picks up the pace of the exploration and delves into the algebraic theory that governs systems of linear equations. The goal is to build intuition and eventually master the art of hunting down the values of variables!

For example, what does your intuition tell you about these two balances? If all blue squares are one weight, and all red circles are another, is the situation above possible?

In addition to balances, this chapter will use everything from graphs to arrays of shapes to combination locks to explore the concept of equations and variables thoroughly!

For example, in the array puzzles in this chapter, each symbol acts the same way as a variable in an equation. The number next to each row or column represents the sum of the values in that row or column.

What is the value of the red star in the array above?

Even if they have many variables, questions like this one can be solved extremely quickly if you know the right techniques!

What is $$d \, ?$$

\begin{align} a + b + c + d &= 8\\ a + b + c + e &= 12\\ a + b + d + e &= 16\\ a + c + d + e &= 20\\ b + c + d + e &= 24 \end{align}

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