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Determine the set [tex]R[/tex] defined by the equation:

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


Sagot :

Certainly! Let's solve the equation step by step:

We are given the equation:
[tex]\[ 4y - 7 = 0 \][/tex]

To solve for [tex]\( y \)[/tex], we need to isolate [tex]\( y \)[/tex] on one side of the equation. Here are the steps:

1. Rewrite the equation to make it clearer:
[tex]\[ 4y - 7 = 0 \][/tex]

2. Add 7 to both sides of the equation to move the constant term to the right side.
[tex]\[ 4y - 7 + 7 = 0 + 7 \][/tex]
Simplifying this, we get:
[tex]\[ 4y = 7 \][/tex]

3. Divide both sides of the equation by 4 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{7}{4} \][/tex]

So, the solution to the equation [tex]\( 4y - 7 = 0 \)[/tex] is:
[tex]\[ y = \frac{7}{4} \][/tex]

This means that in the real coordinate plane [tex]\( \mathbb{R}^2 \)[/tex], the set of all points [tex]\((x, y)\)[/tex] that satisfy the equation [tex]\( 4y - 7 = 0 \)[/tex] consists of all points where the [tex]\( y \)[/tex]-coordinate is [tex]\( \frac{7}{4} \)[/tex]. Therefore, the set can be described as:
[tex]\[ \operatorname{R}\left\{(x, y) \in \mathbb{R}^2 \mid y = \frac{7}{4} \right\} \][/tex]

This represents a horizontal line in the coordinate plane where [tex]\( y \)[/tex] is always [tex]\( \frac{7}{4} \)[/tex], and [tex]\( x \)[/tex] can be any real number.