Question

Lines 𝑥 x and 𝑦 y are parallel. If 𝐵 = 7 𝑥 + 16 , 𝐹 = 3 𝑥 + 28 B=7x+16, F=3x+28 and 𝐷 = 2 𝑦 + 18 D=2y+18 then find the values of 𝑥 x and 𝑦 y . The figure is not drawn to scale.

Answer 1

y is wrong

Answer 2

To solve the problem, we first note that since lines \( x \) and \( y \) are parallel, we can use properties of angles formed by a transversal intersecting parallel lines.

From the information given:

- \( B = 7x + 16 \)
- \( F = 3x + 28 \)
- \( D = 2y + 18 \)

Assuming \( B \) and \( F \) are corresponding angles or alternate interior angles, we set them equal to each other (since angles corresponding to parallel lines and a transversal are equal):

\[
7x + 16 = 3x + 28
\]

Let's solve for \( x \):

1. Subtract \( 3x \) from both sides:
\[
7x - 3x + 16 = 28
\]
\[
4x + 16 = 28
\]

2. Subtract 16 from both sides:
\[
4x = 28 - 16
\]
\[
4x = 12
\]

3. Divide by 4:
\[
x = 3
\]

With \( x \) found, we substitute \( x = 3 \) into \( D \) in order to determine \( y \). However, we actually need to determine a relation for \( y \). Since \( D \) is equal to either \( B \) or \( F \) (again assuming some relationship here), we would also expect that \( D = 2y + 18 = 7x + 16 \) or \( D = 2y + 18 = 3x + 28 \).

Let’s use \( B \) since it involves \( 7x + 16 \):

1. Substitute \( x = 3 \) back into \( B \):
\[
B = 7(3) + 16 = 21 + 16 = 37
\]

So we set \( D \) equal to \( B \):

\[
2y + 18 = 37
\]

2. Subtract 18 from both sides:
\[
2y = 37 - 18
\]
\[
2y = 19
\]

3. Divide by 2:
\[
y = \frac{19}{2} = 9.5
\]

Finally, the solution gives us:
- \( x = 3 \)
- \( y = 9.5 \)

Thus, the values obtained are \( x = 3 \) and \( y = 9.5 \).

Answer 3

Let's clarify the relationships between the angles and ensure the calculations for \( y \) are correct.

Given the equations:

- \( B = 7x + 16 \)
- \( F = 3x + 28 \)
- \( D = 2y + 18 \)

We previously determined \( x \) correctly. Let's first find \( x \) again for confirmation and then check the \( y \) calculation.

### Step 1: Solve for \( x \)
Setting \( B = F \):
\[
7x + 16 = 3x + 28
\]

Subtract \( 3x \) from both sides:
\[
4x + 16 = 28
\]

Subtract \( 16 \) from both sides:
\[
4x = 12
\]

Divide by \( 4 \):
\[
x = 3
\]

### Step 2: Substitute \( x \) into \( B \) and solve for \( D \)

Now substituting \( x = 3 \) into \( B \):
\[
B = 7(3) + 16 = 21 + 16 = 37
\]

### Step 3: Set \( D \) equal to \( B \) (since they are corresponding angles)
\[
D = 2y + 18 = B
\]
Thus:
\[
2y + 18 = 37
\]

### Step 4: Solve for \( y \)
Subtract 18 from both sides:
\[
2y = 37 - 18 = 19
\]

Divide by 2:
\[
y = \frac{19}{2} = 9.5
\]

**It appears that the calculation was performed correctly based on the relationships assumed. The angles \( B \), \( F \), and \( D \) were analyzed correctly with \( B \) and \( D \) assumed to be the corresponding angles**, but if \( D \) corresponds to a different angle, we would need to know that relationship.

### Additional Check: Confirm Relationship Between \( D \) and \( B/F \)
If the relationship between \( D \) and \( F \) was meant to be analyzed or there's a particular angle relationship not mentioned, please clarify that relationship.

Given the current assumptions:
- \( x = 3 \)
- \( y = 9.5 \)

If you have specific conditions or details that need to be factored in for the angle relationships, please provide them, and I will gladly adjust the calculations!