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Problem 589

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

Expert verified
The simultaneous solution set of the given system of equations is $$\{(-2, 5), (-6, -3)\}$$.
See the step by step solution

## 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.

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

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