Problem 17 Solve each system. $$\begin{ar... [FREE SOLUTION]

Chapter 14: Problem 17

Solve each system. $$\begin{array}{l} 3 x^{2}+4 y=-1 \\ x^{2}+3 y=-12 \end{array}$$

Short Answer

Expert verified

To solve the given system of nonlinear equations, first solve the second equation for x in terms of y: $x = \pm\sqrt{-3y - 12}$. Substitute this expression into the first equation and solve for y, finding $y = 7$. Next, substitute y back into the expression for x to find the corresponding values of x: $x = \pm\sqrt{-33}$. The solution to the system is (x, y) = $\left(\sqrt{-33}, 7\right), \left(-\sqrt{-33}, 7\right)$.

Step by step solution

Solve the second equation for x

Express x in terms of y from the second equation: $x^2 = -3y - 12$ $x = \pm\sqrt{-3y - 12}$ We'll consider both positive and negative square roots, as both can lead to different solutions.

Substitute the expression for x in the first equation and solve for y

Substitute the expression for x (positive square root) in the first equation: $3(\sqrt{-3y - 12})^{2} + 4y = -1 $ This simplifies to: $3(-3y-12) + 4y = -1$ Now solve for y: $-9y-36 +4y = -1$ $-5y=-35$ $y = 7$ Now substitute the expression for x (negative square root) in the first equation: $3(-\sqrt{-3y - 12})^{2} + 4y = -1 $ This simplifies to the same equation as before (since squaring eliminates the negative sign), so we get the same result: $y = 7$

Find the corresponding values of x

Substitute the obtained value of y into the expression for x: $x = \pm\sqrt{-3(7) - 12}$ $x = \pm\sqrt{-21 - 12}$ $x = \pm\sqrt{-33}$ Therefore, the solution to the given system is: (x, y) = $\left(\sqrt{-33}, 7\right), \left(-\sqrt{-33}, 7\right)$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations

Quadratic equations form the foundation of many algebraic problems and usually appear in the format: $ ax^2 + bx + c = 0 $. Here, the equation involves the variable $ x $ raised to its second power, making it a quadratic. In our original system, the term $ 3x^2 $ is part of the first equation, indicating it's quadratic in nature. Quadratics can be solved using various methods such as factoring, using the quadratic formula, or completing the square. In this problem, we initially solve one equation for $ x $, which involves quadratic terms. This involves isolating terms and applying further algebraic techniques to find $ x $ values.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for a variable, then substituting that expression into another equation. This converts the system of equations into a solvable form, often reducing it to a single equation. In our problem, we solve for $ x $ from the second equation. Here, $ x = \pm\sqrt{-3y - 12} $ is substituted into the first equation. This allows us to work with a single equation in terms of $ y $, making it easier to solve systematically. Once you find $ y $, you substitute it back into the expression for $ x $ to find corresponding $ x $ values.

Solving Systems Algebraically

Algebraic solutions to systems of equations, like the one presented, involve manipulating equations to find common solutions for all variables. The main goal is to find values that satisfy every equation in the system simultaneously. These systems can be linear, quadratic, or a mix of both as in this case. Such problems are solved by a combination of substitution or elimination methods. The process involves creating expressions and substituting them into one another as seen in our step-by-step solution. The resulting equations are solved for each variable, providing a set of solutions that work for all equations at once.

Complex Numbers

Complex numbers come into play when the square root of a negative number is involved, introducing an imaginary unit $ i $, where $ i^2 = -1 $. In our problem, after finding $ y = 7 $, we extract $ x = \pm\sqrt{-33} $. Since $ -33 $ is negative, $ \sqrt{-33} $ involves a complex number. Rewriting gives $ x = \pm i \sqrt{33} $. Complex numbers, written as $ a + bi $ where $ a $ and $ b $ are real numbers, expand the solutions to non-real numbers and are crucial in fields like engineering and physics for solving such quadratic equations. They ensure all solutions are accounted for, even when outside the realm of real numbers.

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91影视

Solve each system. $$\begin{array}{l} 3 x^{2}+4 y=-1 \\ x^{2}+3 y=-12 \end{array}$$

Short Answer

Step by step solution

Solve the second equation for x

Substitute the expression for x in the first equation and solve for y

Find the corresponding values of x

Key Concepts

Quadratic Equations

Substitution Method

Solving Systems Algebraically

Complex Numbers

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