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

Find the value of the constant 𝑘, given that 𝑥²𝑦(12𝑥 + 𝑘𝑦) is equivalent to 12𝑥³𝑦 + 2𝑥²𝑦².

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Video Transcript

Find the value of the constant 𝑘, given that 𝑥 squared 𝑦 times 12𝑥 plus 𝑘𝑦 is equivalent to 12𝑥 cubed 𝑦 plus two 𝑥 squared 𝑦 squared.

In this question, we are asked to find the value of a constant 𝑘 using the equivalence of two given algebraic expressions. To find the value of 𝑘, we can start by comparing the two given expressions. We want to compare 12𝑥 cubed 𝑦 plus two 𝑥 squared 𝑦 squared and 𝑥 squared 𝑦 multiplied by 12𝑥 plus 𝑘𝑦. 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 𝑥 squared 𝑦 to obtain 𝑥 squared 𝑦 times 12𝑥 plus 𝑥 squared 𝑦 multiplied by 𝑘𝑦.

We can now simplify each term by recalling that we can multiply variables by adding the exponents. Doing this and evaluating gives us 12𝑥 cubed 𝑦 plus 𝑘𝑥 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 𝑥 squared 𝑦 squared. So, for these expressions to be equivalent, the constant factors must be equal in the two terms. Hence, the value of 𝑘 must be equal to two.