Consider the system of equations below where (-2,-1) represents one solution on the coordinate plane. y = -x² + 3 y = x + 1 What is the (x, y) coordinate of the other solution?
Consider the system of equations below where (-2,-1) represents one solution on the coordinate plane. y = -x² + 3 y = x + 1 What is the (x, y) coordinate of the other solution?
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Question 3**
Consider the system of equations below where \((-2, -1)\) represents one solution on the coordinate plane.
\[
y = -x^2 + 3
\]
\[
y = x + 1
\]
What is the \((x, y)\) coordinate of the other solution?
**Explanation:**
This problem involves solving a system of equations to find the points of intersection. One of the solutions is given as \((-2, -1)\).
1. **Equation 1**: \(y = -x^2 + 3\) represents a downward opening parabola.
2. **Equation 2**: \(y = x + 1\) is a linear equation representing a straight line.
To find the intersection points, set the equations equal to each other:
\[
-x^2 + 3 = x + 1
\]
Rearrange into a standard quadratic equation:
\[
-x^2 - x + 3 - 1 = 0 \quad \Rightarrow \quad -x^2 - x + 2 = 0
\]
Simplify:
\[
-x^2 - x + 2 = 0 \quad \Rightarrow \quad x^2 + x - 2 = 0
\]
Factor the quadratic equation:
\[
(x + 2)(x - 1) = 0
\]
Solutions for \(x\) are:
\[
x = -2 \quad \text{or} \quad x = 1
\]
For \(x = 1\), substitute back into \(y = x + 1\):
\[
y = 1 + 1 = 2
\]
Thus, the other solution is \((1, 2)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78a256bd-ea52-4682-957b-1448c364f8ac%2Fdf07bc88-2404-4d46-855c-e6b0db285e66%2F52bm1d_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 3**
Consider the system of equations below where \((-2, -1)\) represents one solution on the coordinate plane.
\[
y = -x^2 + 3
\]
\[
y = x + 1
\]
What is the \((x, y)\) coordinate of the other solution?
**Explanation:**
This problem involves solving a system of equations to find the points of intersection. One of the solutions is given as \((-2, -1)\).
1. **Equation 1**: \(y = -x^2 + 3\) represents a downward opening parabola.
2. **Equation 2**: \(y = x + 1\) is a linear equation representing a straight line.
To find the intersection points, set the equations equal to each other:
\[
-x^2 + 3 = x + 1
\]
Rearrange into a standard quadratic equation:
\[
-x^2 - x + 3 - 1 = 0 \quad \Rightarrow \quad -x^2 - x + 2 = 0
\]
Simplify:
\[
-x^2 - x + 2 = 0 \quad \Rightarrow \quad x^2 + x - 2 = 0
\]
Factor the quadratic equation:
\[
(x + 2)(x - 1) = 0
\]
Solutions for \(x\) are:
\[
x = -2 \quad \text{or} \quad x = 1
\]
For \(x = 1\), substitute back into \(y = x + 1\):
\[
y = 1 + 1 = 2
\]
Thus, the other solution is \((1, 2)\).
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