3x + 4y = 18 and xy = 6, find the value of 9x2 + 16y2

We are given two equations:

1. \( 3x + 4y = 18 \)
2. \( xy = 6 \)

We need to find the value of \( 9x^2 + 16y^2 \).

### Step 1: Square the first equation

We will first square both sides of the equation \( 3x + 4y = 18 \):

\[
(3x + 4y)^2 = 18^2
\]

Expanding both sides:

\[
(3x)^2 + 2(3x)(4y) + (4y)^2 = 324
\]
\[
9x^2 + 24xy + 16y^2 = 324
\]

### Step 2: Substitute the value of \( xy \)

From the second equation, we know \( xy = 6 \). Substitute this into the above equation:

\[
9x^2 + 24(6) + 16y^2 = 324
\]
\[
9x^2 + 144 + 16y^2 = 324
\]

### Step 3: Solve for \( 9x^2 + 16y^2 \)

Now subtract 144 from both sides:

\[
9x^2 + 16y^2 = 324 - 144
\]
\[
9x^2 + 16y^2 = 180
\]

Thus, the value of \( 9x^2 + 16y^2 \) is \( \boxed{180} \).

Would you like any more details on this solution?

Here are 5 related questions:
1. What are the steps to solve a system of linear equations?
2. How do you approach solving quadratic equations using substitution?
3. What is the geometric interpretation of the system of equations involving \( x \) and \( y \)?
4. How do you apply the method of completing the square in solving quadratic equations?
5. How can you solve systems of nonlinear equations?

**Tip:** Always look for substitution opportunities when you have multiple equations involving the same variables!

Learn how to solve the system of linear and quadratic equations 3x + 4y = 18 and xy = 6. Find the value of 9x^2 + 16y^2 using substitution and algebraic manipulation. This problem involves key concepts in algebra, including linear equations and quadratic formulas. Suitable for students in Grades 9-12.

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