Question Video: Finding Ordered Pairs by Solving Cubic Equations Mathematics • 8th Grade

If (π‘₯Β³ βˆ’ 7, 𝑦 βˆ’ 6) = (1, |βˆ›βˆ’64|), find (𝑦, π‘₯).

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Video Transcript

If the coordinate pair π‘₯ cubed minus seven, 𝑦 minus six is equal to one and the absolute value of the cube root of negative 64, find 𝑦, π‘₯.

In order to calculate the values of π‘₯ and 𝑦, we need to solve two equations, firstly, π‘₯ cubed minus seven is equal to one. Solving this equation, we can calculate the value of π‘₯. We begin by adding seven to both sides of the equation. π‘₯ cubed is equal to eight. As cube rooting is the opposite or inverse of cubing, we can cube root both sides of this equation. π‘₯ is equal to the cube root of eight. The cube root of any positive number must give a positive answer. As two cubed is equal to eight, the cube root of eight must be equal to two.

We can now calculate the value of 𝑦 by solving the equation 𝑦 minus six is equal to the absolute value of the cube root of negative 64. To solve this equation, we will begin by simplifying the right-hand side. The cube root of any negative number gives a negative answer such that the cube root of negative π‘₯ is equal to the negative of the cube root of π‘₯. As four cubed is equal to 64, we know that the cube root of 64 is four. This means that the cube root of negative 64 is negative four.

We need to find the absolute value or modulus of this. The absolute value of negative π‘Ž is equal to π‘Ž. This means that the absolute value of negative four is four, and 𝑦 minus six is equal to four. Adding six to both sides of this equation gives us 𝑦 is equal to 10. Normally, when writing a coordinate or ordered pair, we write the π‘₯-coordinate first. However, in this question, we’re asked to find 𝑦, π‘₯. As 𝑦 is equal to 10 and π‘₯ is equal to two, the correct pair is 10, two.

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