IDNLearn.com: Your trusted source for finding accurate answers. Ask your questions and get detailed, reliable answers from our community of experienced experts.

Rewrite the set notation to clarify the mathematical expression for readability and ensure correct grammar.

[tex]\[ R_2 = \{(x, y) \in \mathbb{R}^2 \mid 4y - 7 = 0\} \][/tex]


Sagot :

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]