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# Algebra

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Find the ordered pair (s, t) that satisfies the system
(s/2) + 5t =3 - 7t + 8s
3t - 6s = 9 - 2t

Jul 30, 2024

#1
+1485
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First, let's simplify both equations to setup an interesting way to solve the problem.

$$(s/2) + 5t = 3 - 7t + 8s → s + 10t = 6 - 14t + 16s → -15s + 24t = 6 \space\space\space \text{(1st equation)} \\ 3t - 6s = 9 -2t → 6s + 5t = 9 \space\space\space \text{(2nd equation)}$$

Now, let's mulitply the 1st equation by 6 and the 2nd equation by 15

These numbers seem random, we when we write it out, we can see why it's so important. We get

$$-90s + 144t = 36 \\ 90s + 75t = 135$$

Now we have the opposite s coefficients. Adding these equations, we get

$$219 t = 171$$

$$t = 171 / 219 = 57 / 73$$

Now, we can find s through the value of t. We get

$$6s + 5(57/73) = 9 \\ 6s + 285/73 = 9 \\ 6s = 9 - 285/73 \\ 6s = 372/73 \\ s = 62 / 73$$

Thus, our ordered pair (s, t) is $$(57/73, 62/73)$$

Thanks! :)

Jul 30, 2024
edited by NotThatSmart  Jul 30, 2024