Question
Question 2 options:
How many solutions does the system have?
y
=
-
2
x
+
1
y
=
-
2
x
2
+
1
How many solutions does the system have?
y
=
-
2
x
+
1
y
=
-
2
x
2
+
1
Answers
GPT-4o mini
To determine how many solutions the given system of equations has, we need to analyze the two equations:
1. \( y = -2x + 1 \)
2. \( y = -2x^2 + 1 \)
**Step 1: Graph the Equations**
- The first equation, \( y = -2x + 1 \), is a straight line with a slope of -2 and a y-intercept of 1.
- The second equation, \( y = -2x^2 + 1 \), is a downward-opening parabola with its vertex at the point \( (0, 1) \).
**Step 2: Set the Equations Equal to Each Other**
To find the intersections of the two equations, we set them equal to each other:
\[
-2x + 1 = -2x^2 + 1
\]
**Step 3: Simplify the Equation**
Subtracting 1 from both sides gives:
\[
-2x = -2x^2
\]
Dividing both sides by -2 (assuming \( x \neq 0 \)) leads to:
\[
x = x^2
\]
Rearranging gives:
\[
x^2 - x = 0
\]
Factoring out \( x \):
\[
x(x - 1) = 0
\]
**Step 4: Find the Solutions for \( x \)**
This gives us the solutions:
\[
x = 0 \quad \text{or} \quad x = 1
\]
**Step 5: Find the Corresponding \( y \)-values**
Now, substituting back into either equation to find the corresponding \( y \)-values:
1. For \( x = 0 \):
\[
y = -2(0) + 1 = 1
\]
2. For \( x = 1 \):
\[
y = -2(1) + 1 = -1
\]
Thus, the points of intersection (solutions to the system) are:
- \( (0, 1) \)
- \( (1, -1) \)
**Conclusion**
The system has 2 solutions.
1. \( y = -2x + 1 \)
2. \( y = -2x^2 + 1 \)
**Step 1: Graph the Equations**
- The first equation, \( y = -2x + 1 \), is a straight line with a slope of -2 and a y-intercept of 1.
- The second equation, \( y = -2x^2 + 1 \), is a downward-opening parabola with its vertex at the point \( (0, 1) \).
**Step 2: Set the Equations Equal to Each Other**
To find the intersections of the two equations, we set them equal to each other:
\[
-2x + 1 = -2x^2 + 1
\]
**Step 3: Simplify the Equation**
Subtracting 1 from both sides gives:
\[
-2x = -2x^2
\]
Dividing both sides by -2 (assuming \( x \neq 0 \)) leads to:
\[
x = x^2
\]
Rearranging gives:
\[
x^2 - x = 0
\]
Factoring out \( x \):
\[
x(x - 1) = 0
\]
**Step 4: Find the Solutions for \( x \)**
This gives us the solutions:
\[
x = 0 \quad \text{or} \quad x = 1
\]
**Step 5: Find the Corresponding \( y \)-values**
Now, substituting back into either equation to find the corresponding \( y \)-values:
1. For \( x = 0 \):
\[
y = -2(0) + 1 = 1
\]
2. For \( x = 1 \):
\[
y = -2(1) + 1 = -1
\]
Thus, the points of intersection (solutions to the system) are:
- \( (0, 1) \)
- \( (1, -1) \)
**Conclusion**
The system has 2 solutions.