Lucid1223
Answered

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Given the system of equations:

[tex]\[
\begin{array}{l}
4x - 9y = 7 \\
-4x + 9y = -7
\end{array}
\][/tex]

Which of the following is true?

A. The equations represent the same line, so there is no solution.

B. The equations represent the same line, so there are infinite solutions.

C. The equations represent parallel lines, so there are infinite solutions.

D. The equations represent parallel lines, so there is no solution.

Sagot :

GinnyAnswer
Let's analyze the given system of linear equations step by step:

[tex]\[ \begin{array}{l} 4x - 9y = 7 \quad \text{(1)} \\ -4x + 9y = -7 \quad \text{(2)} \end{array} \][/tex]

### Step 1: Identify the coefficients

Compare the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in both equations.

For equation (1):
- Coefficient of [tex]\(x\)[/tex] is [tex]\(a_1 = 4\)[/tex]
- Coefficient of [tex]\(y\)[/tex] is [tex]\(b_1 = -9\)[/tex]
- Constant term is [tex]\(c_1 = 7\)[/tex]

For equation (2):
- Coefficient of [tex]\(x\)[/tex] is [tex]\(a_2 = -4\)[/tex]
- Coefficient of [tex]\(y\)[/tex] is [tex]\(b_2 = 9\)[/tex]
- Constant term is [tex]\(c_2 = -7\)[/tex]

### Step 2: Check the ratios of the coefficients

We compare the ratios of the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex] and the constant terms [tex]\(c\)[/tex].

1. Ratio of [tex]\(x\)[/tex] coefficients:
[tex]\[ \frac{a_1}{a_2} = \frac{4}{-4} = -1 \][/tex]

2. Ratio of [tex]\(y\)[/tex] coefficients:
[tex]\[ \frac{b_1}{b_2} = \frac{-9}{9} = -1 \][/tex]

3. Ratio of constant terms:
[tex]\[ \frac{c_1}{c_2} = \frac{7}{-7} = -1 \][/tex]

### Step 3: Compare the ratios

Since all ratios [tex]\(\frac{a_1}{a_2}\)[/tex], [tex]\(\frac{b_1}{b_2}\)[/tex], and [tex]\(\frac{c_1}{c_2}\)[/tex] are equal:

[tex]\[ \frac{4}{-4} = \frac{-9}{9} = \frac{7}{-7} = -1 \][/tex]

Thus, the ratios of the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and the constant terms are all equal.

### Conclusion

Since the ratios of the coefficients and the constant terms are equal, it implies that the two equations represent the same line. Therefore, the system of equations has infinite solutions.

Answer: The equations represent the same line, so there are infinite solutions.
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