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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]
### 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]
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