Question
Which equation represents the solution to this system of inequalities?
−3x+y≥−4
y<−2x+3
HINT:
(1 point)
Responses
−3x+y≥−4
y<−2x+3
HINT:
(1 point)
Responses
Answers
Answered by
GPT-4o mini
To find the equations that represent the solution to the given system of inequalities, we will convert the inequalities into equations.
1. **First inequality:**
\(-3x + y \geq -4\)
This can be rearranged into the equation of the line:
\(y = 3x - 4\) (by adding \(3x\) to both sides).
2. **Second inequality:**
\(y < -2x + 3\)
This can also be expressed as an equation:
\(y = -2x + 3\).
Now we can summarize the two lines as:
- The line from the first inequality: \(y = 3x - 4\)
- The line from the second inequality: \(y = -2x + 3\)
These two equations represent lines that divide the coordinate plane into different regions. The solution to the system of inequalities will be the intersection region that satisfies both inequalities simultaneously.
To visualize this:
- The area above or on the line \(y = 3x - 4\) (due to \(\geq\)).
- The area below the line \(y = -2x + 3\) (due to \(<\)).
Thus, the equations that represent the borders of the solution region for the system of inequalities are:
1. \(y = 3x - 4\)
2. \(y = -2x + 3\)
### The equations are:
1. \(y = 3x - 4\)
2. \(y = -2x + 3\)
Select any of these two equations depending on your options. If asked to choose only one, you can select either of them as a representative equation of the solution of the system.
1. **First inequality:**
\(-3x + y \geq -4\)
This can be rearranged into the equation of the line:
\(y = 3x - 4\) (by adding \(3x\) to both sides).
2. **Second inequality:**
\(y < -2x + 3\)
This can also be expressed as an equation:
\(y = -2x + 3\).
Now we can summarize the two lines as:
- The line from the first inequality: \(y = 3x - 4\)
- The line from the second inequality: \(y = -2x + 3\)
These two equations represent lines that divide the coordinate plane into different regions. The solution to the system of inequalities will be the intersection region that satisfies both inequalities simultaneously.
To visualize this:
- The area above or on the line \(y = 3x - 4\) (due to \(\geq\)).
- The area below the line \(y = -2x + 3\) (due to \(<\)).
Thus, the equations that represent the borders of the solution region for the system of inequalities are:
1. \(y = 3x - 4\)
2. \(y = -2x + 3\)
### The equations are:
1. \(y = 3x - 4\)
2. \(y = -2x + 3\)
Select any of these two equations depending on your options. If asked to choose only one, you can select either of them as a representative equation of the solution of the system.