Use substitution to determine whether the given ordered pairs are solutions of the given equation.

left parenthesis negative one seventh comma negative two thirds right parenthesis−17,−23​, 

left parenthesis 0 comma five thirds right parenthesis0,53​; 

77aplus+33bequals=55

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Part 1

Is the ordered pair 

left parenthesis negative one seventh comma negative two thirds right parenthesis−17,−23 

a solution of the equation 

77aplus+33bequals=55​? 

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

No because after substituting for a and​ b, it gives the result 

enter your response herenot equals≠55.

​(Type an integer or a simplified​ fraction.)

B.

Yes because after substituting for a and​ b, it gives the result 

55equals=55.

To determine whether the ordered pair \(\left(-\frac{1}{7}, -\frac{2}{3}\right)\) is a solution to the equation \(7a + 3b = 5\), we'll substitute \(a = -\frac{1}{7}\) and \(b = -\frac{2}{3}\) into the equation and then simplify.

### Substitution and Simplification:

1. **Substitute \(a = -\frac{1}{7}\) and \(b = -\frac{2}{3}\) into the equation:**

   \[
   7\left(-\frac{1}{7}\right) + 3\left(-\frac{2}{3}\right) = 5
   \]

2. **Simplify each term:**

   - For the first term: 
     \[
     7 \times -\frac{1}{7} = -1
     \]
   
   - For the second term:
     \[
     3 \times -\frac{2}{3} = -2
     \]

3. **Add the results:**

   \[
   -1 + (-2) = -3
   \]

Now, we compare the result to the right side of the equation:

\[
-3 \neq 5
\]

### Conclusion:

The left side of the equation \(-3\) does not equal the right side \(5\). Therefore, the ordered pair \(\left(-\frac{1}{7}, -\frac{2}{3}\right)\) is **not** a solution to the equation.

### Correct Answer:

**A. No, because after substituting for \(a\) and \(b\), it gives the result \(-3 \neq 5\).**

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Here are five follow-up questions to deepen your understanding:

1. What would happen if the ordered pair was \(\left(\frac{1}{7}, \frac{2}{3}\right)\)? Would it be a solution?
2. How would the solution change if the equation were \(7a + 3b = -5\)?
3. Is there a different ordered pair that could satisfy the equation \(7a + 3b = 5\)?
4. What does it mean graphically if an ordered pair is a solution to the equation?
5. How can you determine if a pair of fractions is a solution without performing calculations?

**Tip**: When dealing with fractions, double-check your simplifications to avoid errors in the calculation process.

Understand how to determine if given ordered pairs are solutions to equations using substitution. Explore examples and learn the process step-by-step. Suitable for Grades 7-9 mathematics students.

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