https://file.aiquickdraw.com/mathbot/file/1730401781535bkm2mgw3.jpg

To solve for \( p \) and \( q \) so that \( x = 6 \) is the only solution to the equation:

\[
2(px - 7) = 6x + q
\]

we need to expand the left side and compare coefficients.

### Step 1: Expand the Left Side
Expanding \( 2(px - 7) \) gives:
\[
2(px - 7) = 2px - 14
\]

So, the equation now becomes:
\[
2px - 14 = 6x + q
\]

### Step 2: Set Up Conditions for a Unique Solution at \( x = 6 \)
For \( x = 6 \) to be the only solution, the coefficients of \( x \) on both sides must match, and the constant terms must also match. This gives us a system of equations:

1. **Matching the coefficients of \( x \):**
   \[
   2p = 6
   \]

2. **Matching the constant terms:**
   \[
   -14 = q
   \]

### Step 3: Solve for \( p \) and \( q \)

From the first equation:
\[
2p = 6 \implies p = \frac{6}{2} = 3
\]

From the second equation:
\[
q = -14
\]

### Solution
The values of \( p \) and \( q \) that make \( x = 6 \) the only solution are:
\[
p = 3 \quad \text{and} \quad q = -14
\]

Would you like further explanation on this solution or have any questions?

---

Here are some related questions to expand on this concept:

1. How would you solve this equation if you needed a different value of \( x \) as the only solution?
2. What if the equation had been \( 2(px - 7) = 8x + q \)? How would that change the solution for \( p \) and \( q \)?
3. How do you determine if an equation has no solution, one solution, or infinitely many solutions?
4. Can you explain why matching coefficients helps us find unique values for parameters like \( p \) and \( q \)?
5. What happens if \( q \) is a function of \( x \), for example, \( q = x + 5 \)?

**Tip:** When solving equations with parameters, always check if you can equate coefficients to simplify finding unknown values.

Find values for p and q so that x = 6 is the only solution to the equation.

2(px - 7) = 6x + q

p =
q =

Learn how to determine values for p and q so that x = 6 is the only solution to the equation 2(px - 7) = 6x + q. This solution involves expanding, comparing coefficients, and solving a system of equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation