To solve the system of linear equations
[tex]\[
\begin{cases}
12x = 4y - 4 \\
y = 3x + 1
\end{cases}
\][/tex]
we will proceed step by step to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Substitute the second equation into the first:
The second equation gives us [tex]\( y = 3x + 1 \)[/tex]. We can substitute [tex]\( y = 3x + 1 \)[/tex] into the first equation to solve for [tex]\( x \)[/tex].
[tex]\[
12x = 4(3x + 1) - 4
\][/tex]
2. Simplify the equation:
Distribute the 4 on the right-hand side:
[tex]\[
12x = 12x + 4 - 4
\][/tex]
This simplifies to:
[tex]\[
12x = 12x
\][/tex]
This equation is always true for any value of [tex]\( x \)[/tex].
3. Interpret the solution:
Since the simplified equation [tex]\( 12x = 12x \)[/tex] is always true, this indicates that the system of equations is dependent, meaning there are infinitely many solutions.
To express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], we use the second original equation:
[tex]\[
y = 3x + 1
\][/tex]
4. Individual solution expression:
Since [tex]\( x \)[/tex] can take any value, [tex]\( y = 3x + 1 \)[/tex] will give us the corresponding value of [tex]\( y \)[/tex].
To check consistency with the given answer:
The relation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be rewritten from [tex]\( y = 3x + 1 \)[/tex] as:
[tex]\[
x = \frac{y - 1}{3}
\][/tex]
Thus, the relationship [tex]\( x = \frac{y}{3} - \frac{1}{3} \)[/tex] confirms that there are infinitely many solutions corresponding to the relation [tex]\( x = \frac{y}{3} - \frac{1}{3} \)[/tex].
So, the system of equations
[tex]\[
\begin{cases}
12x = 4y - 4 \\
y = 3x + 1
\end{cases}
\][/tex]
has an infinite number of solutions described by [tex]\( y = 3x + 1 \)[/tex], or equivalently [tex]\( x = \frac{y}{3} - \frac{1}{3} \)[/tex].
