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Lines

𝑥
x and
𝑦
y are parallel. If
𝐶
=
11
𝑥
+
12
,

𝐹
=
7
𝑥
+
24
C=11x+12, F=7x+24 and
𝐸
=
1
𝑦
+
18
E=1y+18 then find the values of
𝑥
x and
𝑦
y . The figure is not drawn to scale

1 answer

To find the values of \( x \) and \( y \), we can start with the given equations for lines \( C \), \( F \), and \( E \).

  1. Understanding the relationships: Since lines \( x \) and \( y \) are parallel, angles formed by transversals crossing parallel lines have certain relationships, particularly corresponding angles are equal.

  2. Given equations: \[ C = 11x + 12 \] \[ F = 7x + 24 \] \[ E = 1y + 18 \]

Since the lines are parallel, we can assume that the measures represented by these equations share some angle relationships. For instance, angle \( C \) can be considered to be equal to angle \( F \) or \( E \).

  1. Finding \( x \): Let's assume that angles \( C \) and \( F \) are equal because they are on the same line, and so we can set up the equation:

\[ 11x + 12 = 7x + 24 \]

Now, we solve for \( x \):

Subtract \( 7x \) from both sides: \[ 11x - 7x + 12 = 24 \] \[ 4x + 12 = 24 \]

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

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

  1. Finding \( y \): Now, substitute \( x = 3 \) back into one of the angles, or set it against \( E \). Let's first calculate \( C \) at \( x = 3 \):

\[ C = 11(3) + 12 = 33 + 12 = 45 \] So, if \( E \) must correspond with the same angle: \[ 1y + 18 = 45 \]

Now, solve for \( y \):

Subtract \( 18 \): \[ 1y = 27 \]

Thus: \[ y = 27 \]

  1. Final answer: The values of \( x \) and \( y \) are: \[ x = 3, \quad y = 27 \]