Question
When graphed on the same coordinate plane, which equation results in a system of equations with exactly one solution?
The line on the coordinate plane has a positive slope and passes through the points (0,-2) , (-6,-4) , (-9,-5)
A. -2x + 6y = -12
B. 3x - 9y = 9
C. x - 3y =6
D. 9x - 3y = 6
The line on the coordinate plane has a positive slope and passes through the points (0,-2) , (-6,-4) , (-9,-5)
A. -2x + 6y = -12
B. 3x - 9y = 9
C. x - 3y =6
D. 9x - 3y = 6
Answers
GPT-4o mini
To determine which equation results in a system of equations with exactly one solution when graphed with the line that has a positive slope and passes through the points (0,-2), (-6,-4), and (-9,-5), we first need to find the equation of the line that these points lie on.
### Step 1: Find the slope of the line
We can find the slope \( m \) using the points (0,-2) and (-6,-4):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-2)}{-6 - 0} = \frac{-4 + 2}{-6} = \frac{-2}{-6} = \frac{1}{3}
\]
So, the slope \( m = \frac{1}{3} \).
### Step 2: Find the equation in point-slope form
Using the point (0, -2) for the y-intercept \( b \):
\[
y = mx + b \implies y = \frac{1}{3}x - 2
\]
### Step 3: Convert to standard form
To convert \( y = \frac{1}{3}x - 2 \) to standard form \( Ax + By = C \), we can multiply everything by 3 to eliminate the fraction:
\[
3y = x - 6 \implies -x + 3y = -6 \implies x - 3y = 6
\]
### Step 4: Analyze the Options
We need to check each option to see if the line represented by the equation will intersect with \( x - 3y = 6 \) at exactly one point.
- **Option A: -2x + 6y = -12**
- Rearranging gives \( 2x - 6y = 12 \) or \( x - 3y = 6 \) (same line as found).
- **Option B: 3x - 9y = 9**
- Rearranging gives \( 3x - 9y = 9 \) or \( x - 3y = 3 \) (different line).
- **Option C: x - 3y = 6**
- This is the same line as the original we derived above.
- **Option D: 9x - 3y = 6**
- Rearranging gives \( 3y = 9x - 6 \) or \( y = 3x - 2 \) (different line).
### Conclusion
The only options that produce a line parallel to the line \( x - 3y = 6 \) is option A, which is also coincident with it, giving us infinitely many solutions. The equation that intersects at exactly one point must not match the slope or intercept in a way that lines are parallel or the same.
The answer is:
**B. 3x - 9y = 9** (which has a different slope).
This option would yield one unique solution along with the original line.
### Step 1: Find the slope of the line
We can find the slope \( m \) using the points (0,-2) and (-6,-4):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-2)}{-6 - 0} = \frac{-4 + 2}{-6} = \frac{-2}{-6} = \frac{1}{3}
\]
So, the slope \( m = \frac{1}{3} \).
### Step 2: Find the equation in point-slope form
Using the point (0, -2) for the y-intercept \( b \):
\[
y = mx + b \implies y = \frac{1}{3}x - 2
\]
### Step 3: Convert to standard form
To convert \( y = \frac{1}{3}x - 2 \) to standard form \( Ax + By = C \), we can multiply everything by 3 to eliminate the fraction:
\[
3y = x - 6 \implies -x + 3y = -6 \implies x - 3y = 6
\]
### Step 4: Analyze the Options
We need to check each option to see if the line represented by the equation will intersect with \( x - 3y = 6 \) at exactly one point.
- **Option A: -2x + 6y = -12**
- Rearranging gives \( 2x - 6y = 12 \) or \( x - 3y = 6 \) (same line as found).
- **Option B: 3x - 9y = 9**
- Rearranging gives \( 3x - 9y = 9 \) or \( x - 3y = 3 \) (different line).
- **Option C: x - 3y = 6**
- This is the same line as the original we derived above.
- **Option D: 9x - 3y = 6**
- Rearranging gives \( 3y = 9x - 6 \) or \( y = 3x - 2 \) (different line).
### Conclusion
The only options that produce a line parallel to the line \( x - 3y = 6 \) is option A, which is also coincident with it, giving us infinitely many solutions. The equation that intersects at exactly one point must not match the slope or intercept in a way that lines are parallel or the same.
The answer is:
**B. 3x - 9y = 9** (which has a different slope).
This option would yield one unique solution along with the original line.