To solve for the set [tex]\(\{(x, y) \in \mathbb{R}^2 \mid 4y - 7 = 0\}\)[/tex], we need to find the values of [tex]\(y\)[/tex] that satisfy the given equation and determine the corresponding [tex]\(x\)[/tex] values, where [tex]\(x\)[/tex] can be any real number.
1. Given Equation:
[tex]\[
4y - 7 = 0
\][/tex]
2. Solve for [tex]\(y\)[/tex]:
[tex]\[
4y - 7 = 0
\][/tex]
To isolate [tex]\(y\)[/tex], add 7 to both sides of the equation:
[tex]\[
4y = 7
\][/tex]
Now, divide both sides by 4:
[tex]\[
y = \frac{7}{4}
\][/tex]
3. Interpret the solution:
The equation [tex]\(4y - 7 = 0\)[/tex] defines a horizontal line in the [tex]\(xy\)[/tex]-plane where [tex]\(y\)[/tex] is always [tex]\(\frac{7}{4}\)[/tex]. This means for any real number [tex]\(x\)[/tex], [tex]\(y\)[/tex] will always be [tex]\(\frac{7}{4}\)[/tex].
4. Form the solution set:
The solution can be expressed as:
[tex]\[
\{(x, y) \in \mathbb{R}^2 \mid y = \frac{7}{4}\}
\][/tex]
Since [tex]\(x\)[/tex] can be any real number, the solution set includes all pairs [tex]\((x, 1.75)\)[/tex], where [tex]\(1.75\)[/tex] is the decimal representation of [tex]\(\frac{7}{4}\)[/tex].
Therefore, the solution set is:
[tex]\[
(x, 1.75) \quad \text{for any } x \in \mathbb{R}
\][/tex]
