Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=1 \\ y+2 x=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has no real solutions.

Step 1: Solve for y in the second equation

Start by isolating \(y\) in the second equation, \(y + 2x = -3\). Subtract \(2x\) from both sides to get \(y = -3 - 2x\).

Step 2: Substitute y in the first equation

Take the expression for \(y\) from Step 1, \(y = -3 - 2x\), and substitute it into the first equation. This gives us \(x^2 + (-3 - 2x)^2 = 1\).

Step 3: Expand the substituted equation

Expand the expression \((-3 - 2x)^2\) to get \(9 + 12x + 4x^2\). The equation becomes \(x^2 + 9 + 12x + 4x^2 = 1\).

Step 4: Simplify the equation

Combine like terms: \(x^2 + 4x^2 = 5x^2\), so the equation simplifies to \(5x^2 + 12x + 9 = 1\). Then, subtract 1 from both sides to get \(5x^2 + 12x + 8 = 0\).

Step 5: Solve the quadratic equation

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5\), \(b = 12\), and \(c = 8\). Calculate the discriminant: \(b^2 - 4ac = 144 - 160 = -16\), which is negative, implying no real solutions for \(x\).

Step 6: Conclude the solution

Since the discriminant is negative, no real values of \(x\) satisfy the equation. Therefore, the system of equations has no real solutions.

Key Concepts

Solving a System of Equations

In mathematics, a system of equations is a set of two or more equations that have the same set of unknowns. Solving these systems means finding the values of the unknowns that satisfy all the equations in the system.
One common method for solving systems of equations is the substitution method. This involves:

• Isolating one of the variables in one of the equations.
• Substituting this expression into the other equation(s).
• Simplifying and solving for the remaining variables.

This method is particularly useful when you have one linear equation and one nonlinear equation, such as a quadratic equation, because it allows you to reduce the number of variables and solve step by step.
For example, in the system given:

• First, solve for \(y\) in the linear equation \(y + 2x = -3\), resulting in \(y = -3 - 2x\).
• Then, substitute this expression into the nonlinear equation \(x^2 + y^2 = 1\).

By substituting \(y\) and simplifying, you transform a system of equations into a single quadratic equation.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\).
To solve such an equation, we use:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the solutions for \(x\), called the roots of the equation. The key steps in applying the quadratic formula include identifying the coefficients \(a\), \(b\), and \(c\) from the equation, and then calculating the discriminant.
In this exercise, the quadratic equation derived from substituting \(y\) is:\(5x^2 + 12x + 8 = 0\).

• Identify the coefficients: \(a = 5\), \(b = 12\), and \(c = 8\).
• Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.

The quadratic formula not only helps in finding the roots but also in understanding if the solutions are real or complex, providing insight into the solution's nature.

The Role of the Discriminant

The discriminant in a quadratic equation is given by the expression \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has one real root (also called a double root).
• If it is negative, the equation has no real roots, implying the solutions are complex numbers.

In our system of equations example, the discriminant is calculated from the quadratic equation \(5x^2 + 12x + 8 = 0\) as:

• \(b^2 - 4ac = 144 - 160 = -16\). This negative value indicates no real solutions exist for \(x\).

Thus, knowing the discriminant provides valuable insight into the potential solutions for a quadratic equation and, consequently, the system of equations.