Problem 46 Solve each system using the elim... [FREE SOLUTION]

Chapter 13: Problem 46

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{r} -2 x^{2}+7 x y-3 y^{2}=4 \\ 2 x^{2}-3 x y+3 y^{2}=4 \end{array} $$

Short Answer

Expert verified

xy = 2 and x = 卤鈭�3.5 with y = 1 (or another suitable substitution).

Step by step solution

- Add the two equations

To eliminate the quadratic terms, add the two given equations: -2x^2 + 7xy - 3y^2 = 4 + 2x^2 - 3xy + 3y^2 = 4 Summing these equations cancels out the x^2 terms: 4xy = 8.

- Solve for xy

Divide both sides of 4xy = 8 by 4: xy = 2.

- Simplify one of the original equations

Substitute xy = 2 into one of the original equations, for example the first: -2x^2 + 7xy - 3y^2 = 4 Using xy = 2: -2x^2 + 7(2) - 3y^2 = 4 - 2x^2 + 14 - 3y^2 = 4 - 2x^2 - 3y^2 = - 10 2x^2 + 3y^2 = 10.

- Solve for x in terms of y (or vice versa)

Simplify the equation from Step 3: 2x^2 + 3y^2 = 10 Substitute a potential value for y, say y = 1: 2x^2 + 3(1)^2 = 10 2x^2 + 3 = 10 2x^2 = 7 x^2 = 3.5 x = 卤鈭�3.5.

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Key Concepts

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

Elimination Method

The elimination method is a popular technique for solving systems of equations. It involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables. In the given exercise, we applied the elimination method to a nonlinear system of equations. This system had quadratic terms, which can make it a bit complex.
First, we added the two equations to cancel out the quadratic terms involving 饾懃虏. This step simplified the problem significantly by reducing it to a linear equation. The simplified equation was then solved for the product 饾懃饾懄.
In summary:

Add or subtract equations to eliminate one variable.
Simplify the result to find the value of one variable or a relation like 饾懃饾懄 = 2.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This technique is straightforward but is mostly useful when dealing with linear equations.
In our provided exercise, once we obtained 饾懃饾懄 = 2 using the elimination method, we substituted this back into one of the original equations. This allowed us to solve for the variables in terms of each other.
Key points to remember about substitution:

Solve for one variable in terms of the other using one equation.
Substitute this expression into the other equation.
Simplify and solve the resulting equation.

Solving Quadratic Equations

Quadratic equations are of the form 饾憥饾懃虏 + 饾憦饾懃 + 饾憪 = 0. Solving them can involve factoring, using the quadratic formula, or completing the square. In our exercise, the equations provided were nonlinear and included quadratic terms like -2饾懃虏 and -3饾懄虏.

To solve such terms, we combined equations to eliminate the quadratic terms, making them easier to handle. Once simplified, we had to solve for 饾懃 and 饾懄 by either factoring or isolating the square root.

Look for ways to factor or use symmetry to simplify quadratic terms.
Apply the quadratic formula or root extraction for isolates squares.

Nonlinear Systems

Nonlinear systems involve equations that are not linear, including terms like 饾懃虏, 饾懄虏, and 饾懃饾懄. They are trickier to solve since they cannot simply be rearranged like linear equations.

Here, our system included terms such as -2饾懃虏 + 7饾懃饾懄 - 3饾懄虏. We used elimination to reduce the complexity of these nonlinear expressions. In some cases, a combination of elimination and substitution methods works best for solving these kinds of equations.

Look for ways to simplify nonlinear terms.
Sometimes you'll need to combine elimination and substitution for the best results.
Ensure to validate your solutions by plugging them back into the original equations.

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91影视

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{r} -2 x^{2}+7 x y-3 y^{2}=4 \\ 2 x^{2}-3 x y+3 y^{2}=4 \end{array} $$

Short Answer

Step by step solution

- Add the two equations

- Solve for xy

- Simplify one of the original equations

- Solve for x in terms of y (or vice versa)

Key Concepts

Elimination Method

Substitution Method

Solving Quadratic Equations

Nonlinear Systems

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