Do the following statement and its converse form a biconditional that is true?

If 

10

k

+

6

=

–

4

,

 then 

k

=

–

1

.


Let's analyze the statement and its converse to determine if they form a biconditional that is true.

### Original Statement:
"If \( 10k + 6 = -4 \), then \( k = -1 \)."

#### Step 1: Solve the original statement
Given the equation \( 10k + 6 = -4 \), we can solve for \( k \):

\[
10k + 6 = -4
\]

Subtract 6 from both sides:

\[
10k = -10
\]

Divide both sides by 10:

\[
k = -1
\]

This shows that the original statement is true.

### Converse Statement:
The converse of the original statement is: "If \( k = -1 \), then \( 10k + 6 = -4 \)."

#### Step 2: Verify the converse
Substitute \( k = -1 \) into the equation \( 10k + 6 \):

\[
10(-1) + 6 = -10 + 6 = -4
\]

The converse statement is also true.

### Biconditional Statement:
A biconditional statement is true if both the original statement and its converse are true. In this case, both the original statement and its converse are true, so the biconditional statement:

\[
10k + 6 = -4 \text{ if and only if } k = -1
\]

is true.

### Conclusion:
Yes, the original statement and its converse form a biconditional that is true.

Would you like any details or have further questions?

---

Here are 5 related questions you might consider:

1. What is the general process for solving linear equations with one variable?
2. How can you determine the truth of a converse statement?
3. What are the conditions under which a biconditional statement is true?
4. How does solving a system of equations differ from solving a single equation?
5. What are some common mistakes to avoid when solving linear equations?

**Tip:** When verifying a converse statement, always substitute the value from the conclusion of the original statement back into the premise to check for consistency.

Explore the concept of biconditional statements in linear equations. Learn how to determine if statements and their converses form true biconditionals with detailed examples. Suitable for high school students studying algebra and logic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation