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Question Video: Finding the Value of a Constant That Makes Two Algebraic Expressions Equivalent Mathematics • 9th Grade

Find the value of the constant π‘˜, given that π‘₯²𝑦(12π‘₯ + π‘˜π‘¦) is equivalent to 12π‘₯³𝑦 + 2π‘₯²𝑦².

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Video Transcript

Find the value of the constant π‘˜, given that π‘₯ squared 𝑦 times 12π‘₯ plus π‘˜π‘¦ is equivalent to 12π‘₯ cubed 𝑦 plus two π‘₯ squared 𝑦 squared.

In this question, we are asked to find the value of a constant π‘˜ using the equivalence of two given algebraic expressions. To find the value of π‘˜, we can start by comparing the two given expressions. We want to compare 12π‘₯ cubed 𝑦 plus two π‘₯ squared 𝑦 squared and π‘₯ squared 𝑦 multiplied by 12π‘₯ plus π‘˜π‘¦. We can simplify the second expression by noting that we have the product of a binomial and a monomial. So we can distribute the monomial factor over every term inside the parentheses. To do this, we multiply each term in the binomial by π‘₯ squared 𝑦 to obtain π‘₯ squared 𝑦 times 12π‘₯ plus π‘₯ squared 𝑦 multiplied by π‘˜π‘¦.

We can now simplify each term by recalling that we can multiply variables by adding the exponents. Doing this and evaluating gives us 12π‘₯ cubed 𝑦 plus π‘˜π‘₯ squared 𝑦 squared.

We can now compare this expression to the other expression we are given. First, we note that the first terms in each of the expressions are identical. Next, in the second term, we note that both terms have the same variables of π‘₯ squared 𝑦 squared. So, for these expressions to be equivalent, the constant factors must be equal in the two terms. Hence, the value of π‘˜ must be equal to two.

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