Chapter 20: Chapter 20

Problem 589

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Obtain the simultaneous solution set of \(x^{2}+2 y^{2}=54\) \(2 \mathrm{x}-\mathrm{y}=-9\)

Short Answer

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The simultaneous solution set of the given system of equations is \(\{(-2, 5), (-6, -3)\}\).

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write down the given equations

The given equations are: 1) \(x^2 + 2y^2 = 54\) 2) \(2x - y = -9\)

Step 2: Solve one equation for one variable

It's generally easier to work with linear equations, so we'll solve equation (2) for y: \(y = 2x + 9\)

Step 3: Substitute the expression for y into the other equation

Substitute \(y = 2x + 9\) into equation (1): \(x^2 + 2(2x + 9)^2 = 54\)

Step 4: Simplify and solve for x

Expanding and simplifying the equation: \(x^2 + 2(4x^2 + 36x + 81) = 54\) \(x^2 + 8x^2 + 72x + 162 = 54\) Now, combine like terms and move everything to one side of the equation to obtain a quadratic equation in x: \(9x^2 + 72x + 108 = 0\) To solve the quadratic equation for x, we can either use factoring, completing the square, or the quadratic formula. Since the coefficients are divisible by 9, let's divide by 9 to make it simpler: \(x^2 + 8x + 12 = 0\) Now, factor the quadratic equation: \((x + 2)(x + 6) = 0\) Set each factor equal to zero and solve for x: \(x + 2 = 0\) => \(x = -2\) \(x + 6 = 0\) => \(x = -6\)

Step 5: Solve for y using the expression for y

Now, substitute the x values into our expression for y: For \(x = -2\): \(y = 2(-2) + 9 = -4 + 9 = 5\) For \(x = -6\): \(y = 2(-6) + 9 = -12 + 9 = -3\)

Step 6: Check the solutions in both equations

Check whether the solutions \((-2, 5)\) and \((-6,-3)\) satisfy both equations: For \((-2, 5)\): 1) \((-2)^2 + 2(5)^2 = 4 + 50 = 54\) 2) \(2(-2) - 5 = -4 - 5 = -9\) For \((-6, -3)\): 1) \((-6)^2 + 2(-3)^2 = 36 + 18 = 54\) 2) \(2(-6) - (-3) = -12 + 3 = -9\) Both solutions satisfy both equations.

Step 7: Final Answer

The simultaneous solution set of the given system of equations is \(\{(-2, 5), (-6, -3)\}\).

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Chapter 20: Chapter 20

Obtain the simultaneous solution set of \(x^{2}+2 y^{2}=54\) \(2 \mathrm{x}-\mathrm{y}=-9\)

Short Answer

Step by step solution

Unlock all solutions

Step 1: Write down the given equations

Step 2: Solve one equation for one variable

Step 3: Substitute the expression for y into the other equation

Step 4: Simplify and solve for x

Step 5: Solve for y using the expression for y

Step 6: Check the solutions in both equations

Step 7: Final Answer

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