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Sagot :
Let's determine the greatest common factor (GCF) for each of the mathematical expressions provided.
### Step-by-Step Solution:
1. Expression A: [tex]\(15x - 6\)[/tex]
- To find the GCF of the terms [tex]\(15x\)[/tex] and [tex]\(-6\)[/tex], we need to identify the largest number that divides both 15 and [tex]\(-6\)[/tex] without a remainder.
- The GCF of 15 and [tex]\(-6\)[/tex] is 3.
- Therefore, the GCF of [tex]\(15x - 6\)[/tex] is 3.
2. Expression B: [tex]\(24x + 18y\)[/tex]
- To find the GCF of the terms [tex]\(24x\)[/tex] and [tex]\(18y\)[/tex], we need to identify the largest number that divides both 24 and 18 without a remainder.
- The GCF of 24 and 18 is 6.
- Therefore, the GCF of [tex]\(24x + 18y\)[/tex] is 6.
3. Expression C: [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex]
- To find the GCF of the terms [tex]\(\frac{3x}{4}\)[/tex] and [tex]\(-\frac{7}{4}xy\)[/tex], we first notice that both terms have a common factor in the denominator and share a factor in the numerator.
- The GCF of the numerators [tex]\(3x\)[/tex] and [tex]\(-7xy\)[/tex] is 3 (ignoring the variables for now).
- Both terms have a common denominator of 4.
- Therefore, the GCF of [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] is [tex]\(\frac{3}{4}\)[/tex] which simplifies to 0.75.
4. Expression D: [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex]
- Simplify the fraction [tex]\(\frac{14}{8}\)[/tex] to [tex]\(\frac{7}{4}\)[/tex].
- Thus, the expression becomes [tex]\(\frac{7}{4}y + \frac{7}{4}y\)[/tex].
- Since both terms are actually the same after simplification, the GCF is [tex]\(\frac{7}{4}\)[/tex].
- Therefore, the GCF of [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] is [tex]\(\frac{7}{4}\)[/tex] which simplifies to 1.75.
### Final Result:
- Expression A: The GCF of [tex]\(15x - 6\)[/tex] is 3.
- Expression B: The GCF of [tex]\(24x + 18y\)[/tex] is 6.
- Expression C: The GCF of [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] is 0.75.
- Expression D: The GCF of [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] is 1.75.
Thus, we have mapped each expression to its respective GCF:
- A. [tex]\(15x - 6\)[/tex] → 3
- B. [tex]\(24x + 18y\)[/tex] → 6
- C. [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] → 0.75
- D. [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] → 1.75
### Step-by-Step Solution:
1. Expression A: [tex]\(15x - 6\)[/tex]
- To find the GCF of the terms [tex]\(15x\)[/tex] and [tex]\(-6\)[/tex], we need to identify the largest number that divides both 15 and [tex]\(-6\)[/tex] without a remainder.
- The GCF of 15 and [tex]\(-6\)[/tex] is 3.
- Therefore, the GCF of [tex]\(15x - 6\)[/tex] is 3.
2. Expression B: [tex]\(24x + 18y\)[/tex]
- To find the GCF of the terms [tex]\(24x\)[/tex] and [tex]\(18y\)[/tex], we need to identify the largest number that divides both 24 and 18 without a remainder.
- The GCF of 24 and 18 is 6.
- Therefore, the GCF of [tex]\(24x + 18y\)[/tex] is 6.
3. Expression C: [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex]
- To find the GCF of the terms [tex]\(\frac{3x}{4}\)[/tex] and [tex]\(-\frac{7}{4}xy\)[/tex], we first notice that both terms have a common factor in the denominator and share a factor in the numerator.
- The GCF of the numerators [tex]\(3x\)[/tex] and [tex]\(-7xy\)[/tex] is 3 (ignoring the variables for now).
- Both terms have a common denominator of 4.
- Therefore, the GCF of [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] is [tex]\(\frac{3}{4}\)[/tex] which simplifies to 0.75.
4. Expression D: [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex]
- Simplify the fraction [tex]\(\frac{14}{8}\)[/tex] to [tex]\(\frac{7}{4}\)[/tex].
- Thus, the expression becomes [tex]\(\frac{7}{4}y + \frac{7}{4}y\)[/tex].
- Since both terms are actually the same after simplification, the GCF is [tex]\(\frac{7}{4}\)[/tex].
- Therefore, the GCF of [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] is [tex]\(\frac{7}{4}\)[/tex] which simplifies to 1.75.
### Final Result:
- Expression A: The GCF of [tex]\(15x - 6\)[/tex] is 3.
- Expression B: The GCF of [tex]\(24x + 18y\)[/tex] is 6.
- Expression C: The GCF of [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] is 0.75.
- Expression D: The GCF of [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] is 1.75.
Thus, we have mapped each expression to its respective GCF:
- A. [tex]\(15x - 6\)[/tex] → 3
- B. [tex]\(24x + 18y\)[/tex] → 6
- C. [tex]\(\frac{3x}{4} - \frac{7}{4}xy\)[/tex] → 0.75
- D. [tex]\(\frac{14}{8}y + \frac{7}{4}y\)[/tex] → 1.75
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