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.
