Question
Consider the system of linear equations 2x + 3y = 8 and 3x + y = –2. Which statement is correct?
The point (1, 2) is not a solution to the system of equations because it satisfies neither equation.
The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.
The point (1, 2) is a solution to the system of equations because it satisfies the equation 2x + 3y = 8.
The point (1, 2) is a solution to the system of equations because it satisfies both equations.
The point (1, 2) is not a solution to the system of equations because it satisfies neither equation.
The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.
The point (1, 2) is a solution to the system of equations because it satisfies the equation 2x + 3y = 8.
The point (1, 2) is a solution to the system of equations because it satisfies both equations.
Answers
GPT-4o mini
To determine which statement is correct, we need to check whether the point (1, 2) satisfies each of the two equations in the system:
1. **Equation 1:** \(2x + 3y = 8\)
- Substitute \(x = 1\) and \(y = 2\):
\[
2(1) + 3(2) = 2 + 6 = 8
\]
This equation is satisfied.
2. **Equation 2:** \(3x + y = -2\)
- Substitute \(x = 1\) and \(y = 2\):
\[
3(1) + 2 = 3 + 2 = 5
\]
This equation is NOT satisfied, since \(5 \neq -2\).
Now that we have checked both equations, we can evaluate the statements:
- **The point (1, 2) is not a solution to the system of equations because it satisfies neither equation.** (This is incorrect; it satisfies the first.)
- **The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.** (This is correct.)
- **The point (1, 2) is a solution to the system of equations because it satisfies the equation 2x + 3y = 8.** (This is incorrect, as the point must satisfy both equations to be considered a solution to the system.)
- **The point (1, 2) is a solution to the system of equations because it satisfies both equations.** (This is incorrect; it does not satisfy the second equation.)
Thus, the correct statement is:
**The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.**
1. **Equation 1:** \(2x + 3y = 8\)
- Substitute \(x = 1\) and \(y = 2\):
\[
2(1) + 3(2) = 2 + 6 = 8
\]
This equation is satisfied.
2. **Equation 2:** \(3x + y = -2\)
- Substitute \(x = 1\) and \(y = 2\):
\[
3(1) + 2 = 3 + 2 = 5
\]
This equation is NOT satisfied, since \(5 \neq -2\).
Now that we have checked both equations, we can evaluate the statements:
- **The point (1, 2) is not a solution to the system of equations because it satisfies neither equation.** (This is incorrect; it satisfies the first.)
- **The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.** (This is correct.)
- **The point (1, 2) is a solution to the system of equations because it satisfies the equation 2x + 3y = 8.** (This is incorrect, as the point must satisfy both equations to be considered a solution to the system.)
- **The point (1, 2) is a solution to the system of equations because it satisfies both equations.** (This is incorrect; it does not satisfy the second equation.)
Thus, the correct statement is:
**The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.**