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Chapter 13: Problem 48

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 4 x^{2}+x y=30 \\ x^{2}+3 x y=-9 \end{array}\right.$$

Short Answer

Expert verified
The solutions of the system of equations are \(y = -6\) and \(x = 9 卤 sqrt(72)\).

Step by step solution

01

Rearrange the second equation

Rearrange the second equation \(x^{2} + 3xy = -9\) to get \(x^{2} = -3xy - 9\). This step helps us to replace \(x^{2}\) in the first equation with the right-hand side of the rearranged version.
02

Substitute into the first equation

Substitute \(x^{2}\) from the rearranged second equation into the first equation \(4x^{2} + xy = 30\) to obtain \[4(-3xy - 9) + xy = 30\]. This step gives a simplified equation in terms of one variable \(y\).
03

Simplify the equation

Simplify the resulting equation, \(-12xy - 36 + xy = 30\), by combining like terms and isolating \(y\), gives us \[-11xy = 66 \] then divide by -11, we get \[y = -6\].
04

Substitute \(y\ into the second equation

Now we substitute \(y=-6\) back into the second equation \(x^{2} + 3xy = -9\), we obtain \[x^{2} + 3x(-6) = -9\]. This simplifies to \(x^{2} - 18x+9 = 0\]. This is now a quadratic formula which we can solve for \(x\).
05

Solve for \(x\)

We can solve the quadratic equation \(x^{2} - 18x+9 = 0\) for \(x\) by applying the quadratic formula \(x = [-b 卤 sqrt(b^{2} - 4ac)]/(2a)\). Substituting the values \(a = 1\), \(b = -18\), and \(c = 9\) into this formula gives us the solutions \(x = 9 卤 sqrt(72)\). Therefore, the solutions are \(x = 9 + sqrt(72)\) and \(x = 9 - sqrt(72)\).

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Most popular questions from this chapter

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=\frac{1}{2}(y+2)^{2}+1$$

How can you distinguish ellipses from circles by looking at their equations?

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$x^{2}+4 y^{2}=16$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=y^{2}+6 y$$

Multiply: \(\quad(3 x-2)\left(2 x^{2}-4 x+3\right)\)
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