Problem 3 - Algebra 2

Question

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{3 x=8 y^{2}} \\ {8 y^{2}-2 x^{2}=16}\end{array} $$

Step-by-Step Solution

Answer

The given system has no real solutions.

Step 1: Express x in terms of y

Start with the first equation: \(3x = 8y^2\). Solve for \(x\) by dividing both sides by 3: \(x = \frac{8}{3}y^2\).

Step 2: Substitute x in the second equation

Take the expression for \(x\) from Step 1 and substitute it into the second equation: \(8y^2 - 2\left(\frac{8}{3}y^2\right)^2 = 16\).

Step 3: Simplify the equation

Simplify the equation from Step 2: \(8y^2 - 2\cdot\frac{64}{9}y^4 = 16\). Multiply through by 9 to clear the fraction: \(72y^2 - 128y^4 = 144\).

Step 4: Rearrange into a polynomial equation

Rearrange the equation from Step 3: \(-128y^4 + 72y^2 - 144 = 0\). Divide the entire equation by \\(-8\) to simplify: \(16y^4 - 9y^2 + 18 = 0\).

Step 5: Let u = y^2

Let \(u = y^2\), which transforms the equation into a quadratic form: \(16u^2 - 9u + 18 = 0\).

Step 6: Solve the quadratic equation

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 16\), \(b = -9\), and \(c = 18\). Calculate the discriminant: \((-9)^2 - 4\cdot16\cdot18 = 81 - 1152 = -1071\). The discriminant is negative, indicating no real solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). It includes a term with a variable raised to the second power. The standard form is characterized by three coefficients: \(a\), \(b\), and \(c\), where \(a eq 0\).Quadratic equations are central in algebra because they can model various real-world situations, like projectile motion and areas.

• The solutions to a quadratic equation are known as the roots and can be real or complex numbers.
• The number of solutions is determined by the equation's discriminant.
• These equations can be solved by factoring, using the quadratic formula, or completing the square.

In our exercise, transforming the system of equations into a quadratic form involves manipulating the variables to see how they interrelate, ultimately leading us to detect any potential solutions.

Discriminant

The discriminant is a key component of the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It is symbolized as \(\Delta\) and is calculated using \(b^2 - 4ac\).Understanding the discriminant helps us predict the nature and number of solutions (or roots) a quadratic equation possesses:

• A positive discriminant \((\Delta > 0)\) suggests two distinct real roots.
• A zero discriminant \((\Delta = 0)\) implies one real root (or a repeated root).
• A negative discriminant \((\Delta < 0)\) indicates that the roots are complex and no real solutions exist.

In this exercise, our transformed equation had a discriminant \(-1071\), highlighting no real solutions. This means that the combination of these specific equations does not intersect on a real plane.

Substitution Method

The substitution method is a straightforward algebraic strategy used to solve systems of equations. It involves solving one equation for a variable and substituting that expression into another equation.Here's how it generally works:

• Take one of the equations and solve for one variable in terms of the others.
• Substitute the resulting expression into the second equation.
• This substitution reduces the system from two variables to a single variable problem.
• Solve the new, simpler equation.
• Back-substitute to find the other variable, if necessary.

In our step-by-step solution, we first expressed \(x\) in terms of \(y\) from the equation \(3x = 8y^2\). By substituting \(x\) into the second equation, we then dealt with a single-variable equation in \(y\). This method is particularly useful for nonlinear systems that include quadratic equations, as seen in this exercise.