Problem 18 - Algebra 2

Question

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}=x^{2}-25} \\ {x^{2}-y^{2}=7}\end{array} $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Identify the given equations

We are provided with two equations:1. \( y^2 = x^2 - 25 \)2. \( x^2 - y^2 = 7 \).These represent a system of quadratic equations that need to be solved simultaneously.

Step 2: Express y² from the second equation

Notice that the second equation can be manipulated to express \( y^2 \) in terms of \( x^2 \):\[ x^2 - y^2 = 7 \Rightarrow y^2 = x^2 - 7 \].

Step 3: Substitute y² in the first equation

Substitute \( y^2 = x^2 - 7 \) from Step 2 into the first equation:\[ x^2 - 7 = x^2 - 25 \].

Step 4: Simplify the new equation

Subtract \( x^2 \) from both sides to simplify:\[ -7 = -25 \].This simplifies further to:\[ 18 = 0 \].This results in a contradiction.

Step 5: Address the contradiction

Since a contradiction arose (\( 18 = 0 \)), this indicates that there is an inconsistency in the system. Thus, the system of equations has no solution.

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be represented in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of systems of equations, a quadratic equation might involve more than one variable, such as \( y^2 = x^2 - 25 \). This is still a quadratic equation because the highest power of the variables is 2.
Quadratic equations can model parabolas when graphed, and their solutions can be found using various methods like factoring, completing the square, or the quadratic formula. In this exercise, the quadratic nature of the equations is crucial because it creates the potential for multiple solutions or different types of interactions between the curves.

Simultaneous Equations

Simultaneous equations are sets of equations containing multiple variables that are solved together. The goal is to find a set of values for these variables that satisfy all of the equations in the system simultaneously. In this exercise, we are dealing with two equations: \( y^2 = x^2 - 25 \) and \( x^2 - y^2 = 7 \).
To solve the system, we manipulate and combine equations to eliminate variables and simplify the process. Typically, substitution or elimination methods are used to find intersections where both equations hold true."solving simultaneously" means finding solutions that work for each equation at the same time.

No Solution

A system of equations is said to have no solution when there are no values for the variables that can satisfy all of the equations. This means that the equations describe curves or lines that do not intersect at any point. In our step-by-step solution, we rearranged the equations only to find that they led to an impossible statement: \( 18 = 0 \).
This is a clear indication that this specific system of equations does not have any points of intersection or solutions. When working with equations, especially non-linear ones like these, such scenarios signal a fundamental inconsistency in the system.

Mathematical Contradiction

A mathematical contradiction arises when two or more statements or equations mutually exclude each other, leading to a logically impossible situation. In the case of this exercise, after simplifying and substituting, we ended up with \( 18 = 0 \). This is a type of contradiction because 18 does never equal 0.
Such contradictions can emerge in hypothetical or incorrect assumptions. In the context of systems of equations, contradictions likely indicate that there's no common solution, as is the case here. Recognizing contradictions helps in understanding the nature of the system being addressed and confirming no solution.