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Fractions and Balances

So far, we've worked with especially nice cases of equations where the values and answers are always whole numbers. It's time to change that - it's time... for fractions!

How many squares balance with one triangle?

What is the weight of one blue square?

Let's look at our last problem from an algebraic perspective.

If a square is $$s$$ and a triangle is $$t,$$ then we can write these equations from the diagram: \begin{align} s + t &= 6 \\ 3s + \frac{1}{2}t &= 8. \end{align} When encountering equations with fractions, a good first step is to eliminate the fractions by multiplying each equation by the least common multiple of the denominators in that equation. For example, if an equation has denominators of 3 and 4, multiply the entire equation by 12.

The only denominator we have is 2, so we are going to multiply the second equation by 2: \begin{align} 2\left(3s + \frac{1}{2}t\right) &= 2\times 8\\ 6s + t &= 16. \end{align} We can move forward from there to solve the equation by either substitution or elimination. Using substitution, we know that $$t = 16 - 6s.$$

Therefore, \begin{align} s + t &= 6 \\ s + (16-6s) &= 6 \\ -5s &= -10 \\ \Rightarrow s &= 2 \\ t &= 4. \end{align}

To eliminate all of the fractions in this equation in one step, we could multiply the equation by any value except which one? $\frac{1}{2}x + \frac{2}{3}y = 7$

What is the weight of one triangle?

\begin{align} \frac {1}{3}x - \frac {2}{5}y &= 10 \\\\ \frac {1}{6}x + \frac {1}{5}y &= -1 \end{align} What is the value of $$x?$$

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