To solve the system of linear equations, we will use substitution or elimination methods. Here's a step-by-step solution:
Given system of equations:
1. [tex]\( 6x = \frac{y}{4} - 5 \)[/tex]
2. [tex]\( 0.24x - 0.01y = -0.2 \)[/tex]
First, let's rearrange the first equation to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
From Equation 1:
[tex]\[ 6x = \frac{y}{4} - 5 \][/tex]
Multiply both sides by 4 to clear the fraction:
[tex]\[ 24x = y - 20 \][/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[ y = 24x + 20 \][/tex]
Next, substitute this expression for [tex]\( y \)[/tex] into Equation 2:
[tex]\[ 0.24x - 0.01(24x + 20) = -0.2 \][/tex]
Distribute [tex]\( -0.01 \)[/tex]:
[tex]\[ 0.24x - 0.24x - 0.2 = -0.2 \][/tex]
When simplified, we get:
[tex]\[ 0 = 0 \][/tex]
This indicates that the two equations are not independent; they are actually the same equation when expressed in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Therefore, the system has infinitely many solutions along the line given by [tex]\( y = 24x + 20 \)[/tex].
Thus, the system is dependent and represents the same line.
The correct answer is:
b. [tex]\(\{(x, y) \mid 24x - y = -20\}\)[/tex]
