Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x^{2}+y^{2}=25 \\ 2 x+y=10\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has two solutions: \( (x, y) = (3,4) \) and \( (x, y) = (-3, 16) \).

Step 1: Isolate a variable

Let's start by solving the second equation for \(y\). Subtract \(2x\) from both sides of \(2x + y = 10\) to get \(y = 10 - 2x\).

Step 2: Substitution

Next, substitute \(y = 10 - 2x\) into the first equation \(x^2 + y^2 = 25\) to obtain \(x^2 + (10 - 2x)^2 = 25\). Expand the bracket and group like terms to form a quadratic equation.

Step 3: Solve the quadratic equation

Solve the quadratic equation to find the values of \(x\). After finding \(x\), substitute its values back into \(y = 10 - 2x\) to find the corresponding values of \(y\).

Step 4: Verify the solution

To check if the solutions are correct, substitute \(x\) and \(y\) back into the original equations and verify if they satisfy the equations.

Key Concepts

System of Equations

A system of equations is essentially a set of two or more equations that are to be solved simultaneously. In the given exercise, we have two equations: \(x^2 + y^2 = 25\) and \(2x + y = 10\). These equations share the same variables, namely \(x\) and \(y\). The goal is to find values for these variables that satisfy both equations at the same time.
When we deal with systems of equations, we have different methods at our disposal to solve them, like substitution, elimination, and using matrices. Here, we focus on the substitution method.
Substitution involves expressing one variable in terms of the other and then substituting it back into the other equation. This method is particularly useful when an equation has already isolated a variable, or when one can easily be manipulated to isolate a variable.

Quadratic Equation

A quadratic equation is a type of polynomial equation that specifically involves a variable raised to the power of two, expressed in the general form \(ax^2 + bx + c = 0\). In this exercise, the quadratic equation arises after substituting one equation into the other, specifically resulting from the equation \(x^2 + (10 - 2x)^2 = 25\).
To solve a quadratic equation, several methods can be used such as factoring, completing the square, and using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Each method has its own advantages and can be chosen based on the form of the quadratic equation. Quadratic equations often have two solutions because they are based on parabolic graphs, which typically intersect the x-axis at two points.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in problem-solving in mathematics. It involves performing operations to rearrange equations and expressions to simplify or solve them. In this context, algebraic manipulation was used to express \(y\) in terms of \(x\) from the equation \(2x + y = 10\), resulting in \(y = 10 - 2x\).
This step is crucial for applying the substitution method. Manipulating equations often involves basic operations like addition, subtraction, multiplication, and division. It also requires expanding expressions, such as \((10 - 2x)^2\), and combining like terms to form a solvable equation.
Through efficient algebraic manipulation, the original system of equations can be reduced into a single equation with one variable, making it easier to find solutions. Developing strong algebraic manipulation skills can significantly enhance one's ability to tackle complex mathematical problems efficiently.

• Break down complex expressions into simpler ones
• Maintain equality by performing operations on both sides of the equation
• Understand and apply rules of exponents and distribution