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Magazine Chapter 9: Problem 13
Solve the given nonlinear system. $$ \left\\{\begin{array}{l} y=2 x-1 \\ y=x^{2} \end{array}\right. $$
Short Answer
Expert verified
The solution to the system is \((x, y) = (1, 1)\).
Step by step solution
01
Set the Equations Equal to Each Other
To solve the system, set the two expressions for \( y \) equal to each other: \( 2x - 1 = x^2 \).
02
Rearrange the Equation
Move all terms to one side of the equation to set it to zero: \( x^2 - 2x + 1 = 0 \).
03
Recognize the Equation Form
Notice that the equation is a perfect square trinomial. It can be rewritten as \( (x-1)^2 = 0 \).
04
Solve for x
Solve \( (x-1)^2 = 0 \) by taking the square root of both sides, giving \( x-1 = 0 \). Thus, \( x = 1 \).
05
Solve for y using x
Substitute \( x = 1 \) back into one of the original equations, say \( y = 2x - 1 \), to find \( y = 2(1) - 1 = 1 \).
06
Verify Solution
Substitute \( x = 1 \) into the second equation \( y = x^2 \) to verify: \( y = 1^2 = 1 \). Thus, the solution \((x, y) = (1, 1)\) satisfies both equations.
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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 fundamental part of algebra. These equations are usually expressed in the form \( ax^2 + bx + c = 0 \). The solutions to these types of equations can be found using a variety of methods such as:
- Factoring: If the equation can be factored into two binomials, the zero-product property can be used to find the solutions.
- Quadratic Formula: This formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), provides solutions to any quadratic equation, whether or not it can be factored.
- Completing the Square: This method involves reshaping the equation into a perfect square trinomial, making it easier to find the solution.
Perfect Square Trinomial
A perfect square trinomial is a special kind of quadratic expression that can be rewritten as the square of a binomial. It takes the form \( (x + a)^2 \) or \( (x - a)^2 \). When expanded, it results in a quadratic like \( x^2 + 2ax + a^2 \) or \( x^2 - 2ax + a^2 \).
- Identification of such expressions simplifies solving since they can be directly transformed into squared forms.
- A perfect square trinomial often makes it evident how the equation's solutions are obtained through "un-squaring."
System of Equations
A system of equations consists of two or more equations with multiple variables that require a single solution that satisfies all equations simultaneously. Such systems can be:
- Linear: Where both equations are linear (straight-line graph), often solved by substitution or elimination.
- Nonlinear: Where at least one equation isn't linear, as in the case of our problem where one equation was quadratic.
Substitution Method
The substitution method is a technique used to find solutions of a system of equations by replacing a variable in one equation with an expression derived from another equation.
- This method often simplifies complex systems by reducing them step-by-step into more manageable forms.
- It's particularly helpful when one equation is already solved for a particular variable.
- After substitution, the system usually reduces to a single equation with one variable, making it easier to solve.
