Problem 31 - Algebra A Combined Function

Question

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} y=\sqrt{x} \\ x^{2}+y^{2}=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((3, \sqrt{3})\).

Step 1: Analyze the System of Equations

We are given two equations: \( y = \sqrt{x} \) and \( x^2 + y^2 = 12 \). The first equation is expressing \( y \) in terms of \( x \), so we can substitute \( y \) in the second equation.

Step 2: Substitute y in Second Equation

Substitute \( y = \sqrt{x} \) into the equation \( x^2 + y^2 = 12 \). This gives us: \( x^2 + (\sqrt{x})^2 = 12 \). Since \( (\sqrt{x})^2 = x \), the equation simplifies to \( x^2 + x = 12 \).

Step 3: Solve the Quadratic Equation

Rearrange the equation \( x^2 + x = 12 \) to form a standard quadratic equation: \( x^2 + x - 12 = 0 \). To solve it use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=1, c=-12 \).

Step 4: Apply the Quadratic Formula

Calculate the discriminant \( b^2 - 4ac \), which is \( 1^2 - 4(1)(-12) = 1 + 48 = 49 \). Since the discriminant is positive, this quadratic equation has two real solutions. The solutions are \( x = \frac{-1 \pm 7}{2} \), which gives \( x = 3 \) and \( x = -4 \). However, since \( y = \sqrt{x} \), \( x \) must be non-negative, hence only \( x = 3 \) is valid.

Step 5: Find Corresponding y Value

Use the valid value of \( x = 3 \) in the equation \( y = \sqrt{x} \): \( y = \sqrt{3} \). So, the solution of the system is \( (3, \sqrt{3}) \).

Key Concepts

Quadratic Equation

Quadratic equations are second-degree algebraic expressions. They usually take the standard form of \( ax^2 + bx + c = 0 \). In our exercise, this can be seen when the terms \( x^2 + x = 12 \) are rearranged to form \( x^2 + x - 12 = 0 \). This is crucial for solving as it allows the application of solving techniques like factoring, completing the square, or using the quadratic formula.
To solve a quadratic equation, you need to find the values of \( x \) that satisfy the equation. The coefficients \( a \), \( b \), and \( c \) determine the specific properties and solutions of the quadratic equation, such as the number of solutions and their nature. In our case:

• \( a = 1 \)
• \( b = 1 \)
• \( c = -12 \)

The discriminant \( b^2 - 4ac \) helps determine whether the equation has two distinct real solutions, one real solution, or no real solutions. In this case, the discriminant is 49, indicating two distinct real solutions. However, the constraints of our problem (\( y = \sqrt{x} \)) limited the valid solution to \( x = 3 \). This highlights how quadratic equations can be solved and interpreted in real-world contexts.

Substitution Method

The substitution method is a useful strategy for solving systems of equations, particularly when one equation is in terms of a single variable. In this exercise, we applied substitution by taking the first equation \( y = \sqrt{x} \) and using it to substitute for \( y \) in the second equation \( x^2 + y^2 = 12 \).
Substitution reduces the system from two equations to one equation with a single variable. Here, by substituting \( y = \sqrt{x} \) into the second equation, we simplified it to \( x^2 + x = 12 \). This is a clear example of how substitution converts a nonlinear system into a simpler quadratic equation, making it easier to solve.
When using the substitution method, a couple of key points to remember are:

• Choose the equation that is easiest to isolate for one variable.
• Apply the substitution accurately to maintain the integrity of the original relationships.
• Solve the resulting equation. Check for all values, but remember to account for any domain restrictions like those with square roots.

This method underscores the power of substitution to manage complex algebraic systems by focusing attention onto single-variable solutions.

Algebraic Expressions

Understanding algebraic expressions is fundamental to solving equations and systems of equations. An algebraic expression consists of constants, variables, and operations such as addition, subtraction, multiplication, and division.
In our exercise, you encounter algebraic expressions like \( y = \sqrt{x} \) and \( x^2 + y^2 = 12 \). These representations allow you to explore relationships between variables. Here, each equation describes how one variable is related to another — either through a direct expression or a combination of operations.
When working with algebraic expressions, it's important to:

• Identify and understand the operations connecting the variables.
• Recognize how changes in one part of the expression affect the whole.
• Simplify expressions where possible. For instance, knowing \( (\sqrt{x})^2 = x \) simplifies our system significantly.

Algebraic manipulation is at the core of solving equations. It allows you to convert complex expressions into more familiar forms such as quadratic equations, which can be solved with a variety of tools. By mastering these expressions, you build a crucial skillset for tackling both straightforward and intricate mathematical problems.