Certainly! Let's go through each item step-by-step to see which items match based on their mathematical forms.
### Step-by-step Matching Process:
1. First, examine each of the given expressions and the ones we need to match them with:
- Expressions to match:
a. [tex]\(4xy^3\)[/tex]
b. [tex]\(14xy\)[/tex]
c. [tex]\(15y^2\)[/tex]
d. -6
e. [tex]\(5x^2\)[/tex]
- Expressions to match to:
1. [tex]\(-12x^2\)[/tex]
2. [tex]\(6xy\)[/tex]
3. 11
4. [tex]\(8xy^3\)[/tex]
5. [tex]\(-3y^2\)[/tex]
2. Match the terms:
- [tex]\(4xy^3\)[/tex]:
- [tex]\(4xy^3\)[/tex] is a term with [tex]\(x\)[/tex] and [tex]\(y\)[/tex] raised to powers.
- We look for a similar term, which is [tex]\(8xy^3\)[/tex].
- Match: [tex]\(4xy^3\)[/tex] matches with [tex]\(8xy^3\)[/tex].
- [tex]\(a\)[/tex] matches with [tex]\(4\)[/tex].
- [tex]\(14xy\)[/tex]:
- [tex]\(14xy\)[/tex] is a term with [tex]\(x\)[/tex] and [tex]\(y\)[/tex] raised to the first power.
- We look for a similar term, which is [tex]\(6xy\)[/tex].
- Match: [tex]\(14xy\)[/tex] matches with [tex]\(6xy\)[/tex].
- [tex]\(b\)[/tex] matches with [tex]\(2\)[/tex].
- [tex]\(15y^2\)[/tex]:
- [tex]\(15y^2\)[/tex] is a term with [tex]\(y\)[/tex] raised to the second power.
- We look for a similar term, which is [tex]\(-3y^2\)[/tex].
- Match: [tex]\(15y^2\)[/tex] matches with [tex]\(-3y^2\)[/tex].
- [tex]\(c\)[/tex] matches with [tex]\(5\)[/tex].
- -6:
- -6 is a constant term with no variables.
- We look for a similar term, which is 11.
- Match: -6 matches with 11.
- [tex]\(d\)[/tex] matches with [tex]\(3\)[/tex].
- [tex]\(5x^2\)[/tex]:
- [tex]\(5x^2\)[/tex] is a term with [tex]\(x\)[/tex] raised to the second power.
- We look for a similar term, which is [tex]\(-12x^2\)[/tex].
- Match: [tex]\(5x^2\)[/tex] matches with [tex]\(-12x^2\)[/tex].
- [tex]\(e\)[/tex] matches with [tex]\(1\)[/tex].
### Final Matches:
- [tex]\(a = 4xy^3\)[/tex] matches with [tex]\(4 = 8xy^3\)[/tex]
- [tex]\(b = 14xy\)[/tex] matches with [tex]\(2 = 6xy\)[/tex]
- [tex]\(c = 15y^2\)[/tex] matches with [tex]\(5 = -3y^2\)[/tex]
- [tex]\(d = -6\)[/tex] matches with [tex]\(3 = 11\)[/tex]
- [tex]\(e = 5x^2\)[/tex] matches with [tex]\(1 = -12x^2\)[/tex]
So, the final matching is as follows:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = 5\)[/tex]
- [tex]\(d = 3\)[/tex]
- [tex]\(e = 1\)[/tex]
Thus, the answer is:
[tex]\[
(4, 2, 5, 3, 1)
\][/tex]
