To solve for $x$ and $y$ in this diagram where lines $m \parallel n$, we need to use the properties of parallel lines and angles formed by a transversal.

Step 1: Identify Angle Relationships

Since $m \parallel n$, the angles formed by the transversal will have the following relationships:

1. Corresponding Angles are equal.
2. Alternate Interior Angles are equal.

Step 2: Set Up Equations Using Corresponding Angles

1. The angle labeled $(4y - 7)^\circ$ is a corresponding angle to $(2x + 3)^\circ$, so: $4y - 7 = 2x + 3$

2. The angle labeled $(6x + 9)^\circ$ is a corresponding angle to $(2x + 3)^\circ$, so: $6x + 9 = 2x + 3$

Step 3: Solve the System of Equations

Let’s solve each equation for $x$ and $y$.

Solve Equation 2 for $x$:

$6x + 9 = 2x + 3$ Subtract $2x$ from both sides: $4x + 9 = 3$ Subtract 9 from both sides: $4x = -6$ Divide by 4: $x = -\frac{3}{2}$

Substitute $x = -\frac{3}{2}$ into Equation 1 to Solve for $y$:

$4y - 7 = 2(-\frac{3}{2}) + 3$ Calculate the right side: $4y - 7 = -3 + 3$ $4y - 7 = 0$ Add 7 to both sides: $4y = 7$ Divide by 4: $y = \frac{7}{4}$

Solution:

$x = -\frac{3}{2}, \quad y = \frac{7}{4}$

Would you like further clarification on any step, or do you have any additional questions?

Follow-up Questions

1. How do alternate interior angles apply to parallel lines and transversals?
2. What other angle relationships are helpful when working with parallel lines?
3. Could we use any other method to verify the values of $x$ and $y$?
4. How would the problem change if lines $m$ and $n$ were not parallel?
5. What is the significance of the transversal in defining angle relationships?

Tip

Always double-check angle types (like corresponding and alternate angles) when working with parallel lines to ensure you set up equations correctly.