Final answer:
The value of x in the equilateral triangle ABC with sides AB=13x+5, BC=17x-19, and AC=8x+35 is x = 6, making the length of each side equal to 83 units.
Step-by-step explanation:
To find the value of x, since ABC is an equilateral triangle, all its sides are equal in length. Therefore, we can set the expressions for the sides equal to each other:
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- AB = 13x + 5
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- BC = 17x - 19
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- AC = 8x + 35
We can create equations by equating AB = BC and then AB = AC.
- 13x + 5 = 17x - 19
- Solve for x: 5 + 19 = 17x - 13x
- x = 24 / 4
- x = 6
Substituting x = 6 into any of the side expressions gives us the length of each side:
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- AB = 13(6) + 5 = 83
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- BC = 17(6) - 19 = 83
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- AC = 8(6) + 35 = 83
The value of x is 6 and the length of each side of the equilateral triangle is 83 units.