Which of the following pairs of values for \(x\) and \(y\) would justify the claim that the two triangles are congruent?

[tex]\[ \frac{6}{2x+3} = \frac{y-4}{\text{unknown value}} \][/tex]

A. \(x = 3\), \(y = 11\)

B. \(x = 5\), \(y = 5\)

C. \(x = 7\), \(y = 9\)

D. [tex]\(x = 9\)[/tex], [tex]\(y = 7\)[/tex]



Answer :

To determine which pair of values for \( x \) and \( y \) makes the expression \( 2x + 3 \) equal to \( y - 4 \), we will evaluate each given pair step by step.

Let's introduce the number pairs one by one and check if the expressions match:

1. Pair 1: \( x = 3 \), \( y = 11 \):
- Calculate \( 2x + 3 \):
\( 2(3) + 3 = 6 + 3 = 9 \)
- Calculate \( y - 4 \):
\( 11 - 4 = 7 \)

The results are:
[tex]\[ 2x + 3 = 9 \quad \text{and} \quad y - 4 = 7 \][/tex]
Since \( 9 \neq 7 \), our condition is not met with this pair.

2. Pair 2: \( x = 5 \), \( y = 5 \):
- Calculate \( 2x + 3 \):
\( 2(5) + 3 = 10 + 3 = 13 \)
- Calculate \( y - 4 \):
\( 5 - 4 = 1 \)

The results are:
[tex]\[ 2x + 3 = 13 \quad \text{and} \quad y - 4 = 1 \][/tex]
Since \( 13 \neq 1 \), our condition is not met with this pair.

3. Pair 3: \( x = 7 \), \( y = 9 \):
- Calculate \( 2x + 3 \):
\( 2(7) + 3 = 14 + 3 = 17 \)
- Calculate \( y - 4 \):
\( 9 - 4 = 5 \)

The results are:
[tex]\[ 2x + 3 = 17 \quad \text{and} \quad y - 4 = 5 \][/tex]
Since \( 17 \neq 5 \), our condition is not met with this pair.

4. Pair 4: \( x = 9 \), \( y = 7 \):
- Calculate \( 2x + 3 \):
\( 2(9) + 3 = 18 + 3 = 21 \)
- Calculate \( y - 4 \):
\( 7 - 4 = 3 \)

The results are:
[tex]\[ 2x + 3 = 21 \quad \text{and} \quad y - 4 = 3 \][/tex]
Since \( 21 \neq 3 \), our condition is not met with this pair.

Since none of the given pairs satisfies the condition [tex]\( 2x + 3 = y - 4 \)[/tex], there is no pair that justifies the claim that the two triangles are congruent based on the provided pairs.