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Magazine Chapter 6: Problem 66
If possible, solve the nonlinear system of equations. $$ \begin{array}{r} x^{2}+x=y \\ 2 x^{2}-y=2 \end{array} $$
Short Answer
Expert verified
The solutions are \((x, y) = (2, 6)\) and \((x, y) = (-1, 0)\).
Step by step solution
01
Express y from the first equation
The first equation of the system is \( x^2 + x = y \). We can express \( y \) in terms of \( x \), which gives \( y = x^2 + x \).
02
Substitute y into the second equation
Substitute \( y = x^2 + x \) into the second equation \( 2x^2 - y = 2 \). This yields: \[ 2x^2 - (x^2 + x) = 2 \]. Simplifying gives \( x^2 - x = 2 \).
03
Simplify the equation
The equation from the previous step is \( x^2 - x = 2 \). Rearrange it to standard quadratic form: \[ x^2 - x - 2 = 0 \].
04
Factor the quadratic equation
Now, factor the quadratic equation \( x^2 - x - 2 = 0 \). It factors into \((x - 2)(x + 1) = 0\).
05
Solve for x
Set each factor equal to zero to solve for \( x \): 1. \( x - 2 = 0 \) gives \( x = 2 \). 2. \( x + 1 = 0 \) gives \( x = -1 \).
06
Find y for each x
Use the equation \( y = x^2 + x \) to find the value of \( y \) for each \( x \):1. If \( x = 2 \), then \( y = 2^2 + 2 = 6 \). 2. If \( x = -1 \), then \( y = (-1)^2 + (-1) = 0 \).
07
Verify solutions
Substitute back into the original equations to verify:1. For \((x, y) = (2, 6)\): - First equation: \( x^2 + x = 4 + 2 = 6 = y \) - Second equation: \( 2x^2 - y = 8 - 6 = 2 \) (Both are satisfied)2. For \((x, y) = (-1, 0)\): - First equation: \( x^2 + x = 1 - 1 = 0 = y \) - Second equation: \( 2x^2 - y = 2 - 0 = 2 \) (Both are satisfied)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
Solving quadratic equations is a vital skill in algebra, and it often appears in dealing with systems of nonlinear equations. A quadratic equation is typically expressed in the standard form: \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The roots of the quadratic equation are solutions for \( x \) that make the left-hand side zero.
Here are a few popular methods to find these solutions:
Here are a few popular methods to find these solutions:
- Factoring: Break down the quadratic into simpler binomial terms, i.e., \((x - p)(x - q) = 0\). The solutions are \( x = p \) and \( x = q \).
- Quadratic Formula: Use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This method is universal and works for any quadratic equation.
- Completing the Square: Rewrite the equation in the form \((x - h)^2 = k\) and solve for \( x \).
Factoring Polynomials
Factoring polynomials involves writing a polynomial as a product of its simplest factors. This operation is useful because it simplifies equations, making them easier to solve, just like in our quadratic equation from the problem: \( x^2 - x - 2 = 0 \).
The key steps in factoring include:
The key steps in factoring include:
- Finding common factors: Look for common terms in each term. For example, in a trivial polynomial like \( ax + ay = a(x + y) \).
- Identifying patterns: Recognize patterns like differences of squares \((a^2 - b^2 = (a+b)(a-b))\) and special polynomials like \( a^2 + 2ab + b^2 = (a+b)^2 \).
- Trinomials: For equations of the form \( x^2 + bx + c \), find two numbers that multiply to \( c \) and add up to \( b \). This was applied directly as \( (x - 2)(x + 1) = 0 \).
Substitution Method
The substitution method is an effective way of solving a system of equations, especially when one equation can easily be solved for a single variable. The method involves replacing a variable in one equation with its equivalent expression from another equation.
Here's a breakdown of the method:
Here's a breakdown of the method:
- Solve one equation for one variable: If a system is \( x^2 + x = y \) and \( 2x^2 - y = 2 \), solve the first equation for \( y \), getting \( y = x^2 + x \).
- Substitute into the other equation: Replace \( y \) in the second equation with \( x^2 + x \) to form a single-variable equation: \( 2x^2 - (x^2 + x) = 2 \).
- Solve the resulting equation: Simplify and solve the equation for the remaining variable, \( x \). Once solved, use the value of \( x \) to find \( y \) using the earlier substitution.
