In Exercises 29–42, solve each system by the method of your choice. {x2+(y−2)2=4x2−2y=0\begin{cases} $$x^2 + (y$$ - $$2)^2 = 4$$ \\ $$x^2 - 2y = 0$$ \end{cases}{x2+(y−2)2=4x2−2y=0

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 37

Chapter 6, Problem 37

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^2 + (y - 2)^2 = 4 $\x$^2 - 2y = 0\(\end{cases}$

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Identify the system of equations to solve: $x^2 + (y - 2)^2 = 4$ and $x^2 - 2y = 0$.

From the second equation, isolate $y$ in terms of $x$: $x^2 - 2y = 0 \implies 2y = x^2 \implies y = \frac{x^2}{2}$.

Substitute the expression for $y$ from step 2 into the first equation: $x^2 + \left( \frac{x^2}{2} - 2 \right)^2 = 4$.

Expand and simplify the equation from step 3 to form a polynomial equation in terms of $x$ only. This will involve squaring the binomial and combining like terms.

Solve the resulting polynomial equation for $x$. Then, substitute each $x$ value back into $y = \frac{x^2}{2}$ to find the corresponding $y$ values, giving the solution points $(x, y)$ for the system.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding all variable values that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, chosen based on equation types and complexity.

Equations of Circles

The equation x² + (y - k)² = r² represents a circle with center at (0, k) and radius r. Understanding this form helps identify geometric constraints and possible solution points when combined with other equations in a system.

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, simplifying the process of finding solutions.

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Textbook Question

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. −2≤x<5

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Textbook Question

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x/4 - y/4 = −1 x + 4y = -9

In Exercises 39–45, graph each inequality. 3x - 4y > 12

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Textbook Question

Write the partial fraction decomposition of each rational expression. $\frac{x^3 + x^2 + 2}{(x^2 + 2)^2}$