Problem 10 - Beginning and Intermediate Algebra

Question

Solve each system. $$\begin{array}{r} y=2 \\ x^{2}+y^{2}=8 \end{array}$$

Step-by-Step Solution

Answer

The solutions to the given system of equations (\(y = 2\) and \(x^2 + y^2 = 8\)) are: 1. \((x, y) = (2, 2)\) 2. \((x, y) = (-2, 2)\)

Step 1: Substitute y in the quadratic equation

We will substitute the value of y from the linear equation (\(y = 2\)) into the quadratic equation (\(x^2 + y^2 = 8\)): \(x^2 + (2)^2 = 8\)

Step 2: Solve for x

Now we will solve the equation \(x^2 + 4 = 8\) for x: \(x^2 = 8 - 4\) \(x^2 = 4\) Taking the square root of both sides: \(x = \pm \sqrt{4}\) so we have: \(x = 2, -2\)

Step 3: Solve for y

Since \(y = 2\) (from the linear equation), we can directly write the values of y corresponding to each value of x: For \(x = 2\), \(y = 2\) For \(x = -2\), \(y = 2\)

Step 4: Write the solution

The solutions to the given system of equations (\(y = 2\) and \(x^2 + y^2 = 8\)) are: 1. \((x, y) = (2, 2)\) 2. \((x, y) = (-2, 2)\)

Key Concepts

Linear Equations

Linear equations are the simplest type of algebraic equations. They involve variables, like \(x\) or \(y\), raised just to the first power. This means there are no exponents involved, making them straightforward to solve.

For example, in the exercise, the linear equation is \(y = 2\). This indicates that no matter the value of \(x\), \(y\) will always be equal to 2. It is represented graphically by a straight line across the Cartesian plane at the constant value of \(y = 2\).

• Linear equations are used to represent relationships with a constant rate of change.
• They often appear in real-life scenarios, like calculating speed or distance with a constant rate.

Understanding linear equations is crucial as they form the basis for more complex equations and systems, providing a foundation for further learning in algebra.

Quadratic Equations

Quadratic equations are algebraic equations of a specific form, containing an unknown variable raised to the second power, typically expressed as \(ax^2 + bx + c = 0\). In the exercise, the equation \(x^2 + y^2 = 8\) is a quadratic equation because of the quadratic term \(x^2\).

These types of equations usually have two solutions because the highest power of the variable \( (x^2) \) indicates the number of possible solutions.

• Quadratic equations can form a parabolic shape when graphed, which helps visualize solutions.
• Solving quadratics can involve factoring, using the quadratic formula, or completing the square.

In our example, the substitution simplifies the quadratic to \(x^2 + 4 = 8\), enabling us to find the solutions for \(x\) more easily by isolating the square term. This leads to finding the values of \(x\) that satisfy the equation.

Substitution Method

The substitution method is a valuable technique for solving systems of equations, particularly when dealing with nonlinear equations like a mix of linear and quadratic equations. This technique involves replacing one variable with an equivalent expression from another equation.

In our example, we substitute \(y = 2\) from the linear equation into the quadratic equation \(x^2 + y^2 = 8\). This step simplifies the system to a single equation in terms of one variable, making it much easier to solve.

• Replacing one variable helps reduce complexity, leading to a more straightforward problem.
• After substituting, it’s crucial to solve the resulting equation accurately to retrieve possible solutions.
• Finally, use the original equation to check solutions by plugging values back in to ensure validity.

This approach not only finds the solutions more quickly but also highlights the essential interplay between different equations in a system.