IDNLearn.com is designed to help you find reliable answers to any question you have. Find accurate and detailed answers to your questions from our experienced and dedicated community members.

Determine the equation of the line that passes through the given points [tex]\((10, 2)\)[/tex] and [tex]\((-3, 2)\)[/tex].

A. [tex]\( y = 2x \)[/tex]
B. [tex]\( y = 2x - 3 \)[/tex]
C. [tex]\( y = 2 \)[/tex]
D. [tex]\( y = 2x + 10 \)[/tex]


Sagot :

To determine the equation of a line that passes through the points [tex]\((10, 2)\)[/tex] and [tex]\((-3, 2)\)[/tex], let's break it down into steps and find the most appropriate answer.

### 1. Identifying the Type of Line

First, examine the coordinates of the points:
- Point 1: [tex]\((10, 2)\)[/tex]
- Point 2: [tex]\((-3, 2)\)[/tex]

Notice that both points have the same y-coordinate of 2. This indicates that the line is horizontal, since the change in the y-coordinates ([tex]\(\Delta y\)[/tex]) is 0.

### 2. Determining the Slope

The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting in our points [tex]\((10, 2)\)[/tex] and [tex]\((-3, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 2}{-3 - 10} = \frac{0}{-13} = 0 \][/tex]

Thus, the slope [tex]\(m\)[/tex] is 0, confirming that the line is horizontal.

### 3. Equation of a Horizontal Line

A horizontal line has a constant y-value for all x-coordinates. The general equation of a horizontal line is:
[tex]\[ y = \text{constant} \][/tex]

Since the y-coordinate of both points is 2, the equation of the line is:
[tex]\[ y = 2 \][/tex]

### 4. Verifying the Equation

To confirm, we substitute the x-values from both points into the equation [tex]\( y = 2 \)[/tex]:

For point [tex]\((10, 2)\)[/tex]:
[tex]\[ y = 2 \][/tex]
which matches the given y-coordinate.

For point [tex]\((-3, 2)\)[/tex]:
[tex]\[ y = 2 \][/tex]
which also matches the given y-coordinate.

Both points satisfy the equation [tex]\( y = 2 \)[/tex].

### 5. Choosing the Correct Option

Given the options:
a. [tex]\( y = 2x \)[/tex]
b. [tex]\( y = 2x - 3 \)[/tex]
c. [tex]\( y = 2 \)[/tex]
d. [tex]\( y = 2x + 10 \)[/tex]

The correct equation of the line that passes through the points [tex]\((10, 2)\)[/tex] and [tex]\((-3, 2)\)[/tex] is:
[tex]\[ \boxed{y = 2} \][/tex]