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.