Certainly! Let’s solve this system of linear equations step-by-step:
Given the system:
1. [tex]\(70y = -77x\)[/tex]
2. [tex]\(-88x - 80y = 0\)[/tex]
### Step 1: Express One Variable in Terms of the Other
First, we solve equation (1) for [tex]\(y\)[/tex]:
[tex]\[ 70y = -77x \][/tex]
[tex]\[ y = \frac{-77}{70}x \][/tex]
[tex]\[ y = \frac{-11}{10}x \][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] in the Second Equation
Next, we substitute [tex]\(y = \frac{-11}{10}x\)[/tex] into the second equation:
[tex]\[ -88x - 80y = 0 \][/tex]
[tex]\[ -88x - 80\left(\frac{-11}{10}x\right) = 0 \][/tex]
[tex]\[ -88x + 88x = 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
This identity confirms that our expression [tex]\(y = \frac{-11}{10}x\)[/tex] is correct and that the system has infinitely many solutions. The relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be written as:
[tex]\[ y = \frac{-11}{10}x \][/tex]
or
[tex]\[ x = -\frac{10}{11}y \][/tex]
### Conclusion
The solution to the system of equations is given by the relationship:
[tex]\[ \boxed{x = -\frac{10}{11}y} \][/tex]
This indicates that [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are not independently determined, but are linearly dependent on each other by the ratio [tex]\(-\frac{10}{11}\)[/tex]. Any value of [tex]\(y\)[/tex] will yield a corresponding value of [tex]\(x\)[/tex] via this relationship.
