Question
The graphed line shown below is y = negative 3 x + 6.
On a coordinate plane, a line goes through points (1, 3) and (2, 0).
Which equation, when graphed with the given equation, will form a system that has no solution?
y = 3 x + 6
y = negative 3 (x + 6)
y = negative 3 (x minus 2)
y = 3 (x minus 2)
On a coordinate plane, a line goes through points (1, 3) and (2, 0).
Which equation, when graphed with the given equation, will form a system that has no solution?
y = 3 x + 6
y = negative 3 (x + 6)
y = negative 3 (x minus 2)
y = 3 (x minus 2)
Answers
GPT-4o mini
A system of equations has no solution when the lines represented by the equations are parallel to each other. This occurs when the lines have the same slope but different y-intercepts.
The given equation is:
\[ y = -3x + 6 \]
From this equation, we can see that the slope (m) is -3. To find a parallel line, we need another equation that also has a slope of -3 but a different y-intercept.
Let's analyze the options:
1. **\( y = 3x + 6 \)**: This has a slope of 3, which is not parallel.
2. **\( y = -3(x + 6) \)**: This can be rewritten as \( y = -3x - 18 \). It has a slope of -3 but a different y-intercept (-18), making it parallel. Progressing forward, this option could lead to no solution in a system.
3. **\( y = -3(x - 2) \)**: This can be rewritten as \( y = -3x + 6 \), which has the same slope and the same y-intercept. This would give us the same line, so it has infinitely many solutions.
4. **\( y = 3(x - 2) \)**: This can be rewritten as \( y = 3x - 6 \). This has a slope of 3, which is not parallel.
From the above options, the equation that would form a system with no solution is:
**\( y = -3(x + 6) \)**, as it has the same slope (-3) but a different y-intercept (-18).
The given equation is:
\[ y = -3x + 6 \]
From this equation, we can see that the slope (m) is -3. To find a parallel line, we need another equation that also has a slope of -3 but a different y-intercept.
Let's analyze the options:
1. **\( y = 3x + 6 \)**: This has a slope of 3, which is not parallel.
2. **\( y = -3(x + 6) \)**: This can be rewritten as \( y = -3x - 18 \). It has a slope of -3 but a different y-intercept (-18), making it parallel. Progressing forward, this option could lead to no solution in a system.
3. **\( y = -3(x - 2) \)**: This can be rewritten as \( y = -3x + 6 \), which has the same slope and the same y-intercept. This would give us the same line, so it has infinitely many solutions.
4. **\( y = 3(x - 2) \)**: This can be rewritten as \( y = 3x - 6 \). This has a slope of 3, which is not parallel.
From the above options, the equation that would form a system with no solution is:
**\( y = -3(x + 6) \)**, as it has the same slope (-3) but a different y-intercept (-18).