Question
If a line can be represented by the equation y+2=13(x−12) , which of the following sets of points lies on the line?(1 point) Responses (−12,2) and (−9,3) left parenthesis negative 12 comma 2 right parenthesis and left parenthesis negative 9 comma 3 right parenthesis (12,−2) and (13,1) left parenthesis 12 comma negative 2 right parenthesis and left parenthesis 13 comma 1 right parenthesis (−12,2) and (−11,3) left parenthesis negative 12 comma 2 right parenthesis and left parenthesis negative 11 comma 3 right parenthesis (12,−2) and (15,−1)
Answers
Answered by
GPT-4o mini
To determine which set of points lies on the line represented by the equation \( y + 2 = 13(x - 12) \), we'll first transform the equation into the slope-intercept form (y = mx + b).
Starting from the given equation:
\[
y + 2 = 13(x - 12)
\]
We expand the right side:
\[
y + 2 = 13x - 156
\]
Now subtract 2 from both sides:
\[
y = 13x - 158
\]
Now we have the equation of the line in slope-intercept form: \( y = 13x - 158 \).
Next, we will test each of the provided points against this equation to see if they lie on the line.
1. **Point (-12, 2)**:
\[
y = 13(-12) - 158 = -156 - 158 = -314 \quad \text{(not equal to 2)}
\]
2. **Point (-9, 3)**:
\[
y = 13(-9) - 158 = -117 - 158 = -275 \quad \text{(not equal to 3)}
\]
3. **Point (12, -2)**:
\[
y = 13(12) - 158 = 156 - 158 = -2 \quad \text{(equal to -2)}
\]
4. **Point (13, 1)**:
\[
y = 13(13) - 158 = 169 - 158 = 11 \quad \text{(not equal to 1)}
\]
5. **Point (-11, 3)**:
\[
y = 13(-11) - 158 = -143 - 158 = -301 \quad \text{(not equal to 3)}
\]
6. **Point (15, -1)**:
\[
y = 13(15) - 158 = 195 - 158 = 37 \quad \text{(not equal to -1)}
\]
After testing each point:
- From the first option **(-12, 2) and (-9, 3)**: Neither point lies on the line.
- From the second option **(12, -2) and (13, 1)**: Only (12, -2) lies on the line.
- From the third option **(-12, 2) and (-11, 3)**: Neither point lies on the line.
- From the fourth option **(12, -2) and (15, -1)**: Only (12, -2) lies on the line.
Thus, the only point that we found to lie on the line among the options is (12, -2).
Therefore, the correct answer is:
**(12, -2) and (13, 1)** (where (12, -2) lies on the line).
Starting from the given equation:
\[
y + 2 = 13(x - 12)
\]
We expand the right side:
\[
y + 2 = 13x - 156
\]
Now subtract 2 from both sides:
\[
y = 13x - 158
\]
Now we have the equation of the line in slope-intercept form: \( y = 13x - 158 \).
Next, we will test each of the provided points against this equation to see if they lie on the line.
1. **Point (-12, 2)**:
\[
y = 13(-12) - 158 = -156 - 158 = -314 \quad \text{(not equal to 2)}
\]
2. **Point (-9, 3)**:
\[
y = 13(-9) - 158 = -117 - 158 = -275 \quad \text{(not equal to 3)}
\]
3. **Point (12, -2)**:
\[
y = 13(12) - 158 = 156 - 158 = -2 \quad \text{(equal to -2)}
\]
4. **Point (13, 1)**:
\[
y = 13(13) - 158 = 169 - 158 = 11 \quad \text{(not equal to 1)}
\]
5. **Point (-11, 3)**:
\[
y = 13(-11) - 158 = -143 - 158 = -301 \quad \text{(not equal to 3)}
\]
6. **Point (15, -1)**:
\[
y = 13(15) - 158 = 195 - 158 = 37 \quad \text{(not equal to -1)}
\]
After testing each point:
- From the first option **(-12, 2) and (-9, 3)**: Neither point lies on the line.
- From the second option **(12, -2) and (13, 1)**: Only (12, -2) lies on the line.
- From the third option **(-12, 2) and (-11, 3)**: Neither point lies on the line.
- From the fourth option **(12, -2) and (15, -1)**: Only (12, -2) lies on the line.
Thus, the only point that we found to lie on the line among the options is (12, -2).
Therefore, the correct answer is:
**(12, -2) and (13, 1)** (where (12, -2) lies on the line).