In Exercises 19–28, solve each system by the addition method. x^2+y^2=13,x^2−y^2=5

Verified step by step guidance

1. Step 1: The addition method involves adding the two equations together to eliminate one of the variables. In this case, we can add the two equations together to eliminate the y^2 term. The result is $2x^2 = 18$.
2. Step 2: To isolate $x^2$, divide both sides of the equation by 2. This gives us $x^2 = 9$.
3. Step 3: To solve for x, take the square root of both sides. Remember that the square root of a number can be either positive or negative. So, $x = 3$ or $x = -3$.
4. Step 4: Substitute the values of x into either of the original equations to solve for y. For example, if we substitute $x = 3$ into the first equation, we get $y^2 = 13 - 3^2$ or $y^2 = 4$. Similarly, if we substitute $x = -3$, we get $y^2 = 13 - (-3)^2$ or $y^2 = 4$.
5. Step 5: Finally, take the square root of both sides to solve for y. So, $y = 2$ or $y = -2$. Therefore, the solutions to the system of equations are $(3, 2)$, $(3, -2)$, $(-3, 2)$, and $(-3, -2)$.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. In this case, we have a system involving two equations in terms of x and y, which can be solved using various methods, including substitution, elimination, or the addition method.

Addition Method (Elimination Method)

The addition method, also known as the elimination method, is a technique used to solve systems of equations by adding or subtracting the equations to eliminate one variable. This method is particularly useful when the equations are structured in a way that allows for easy cancellation of terms, leading to a simpler equation that can be solved for one variable.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. In the given system, the equations involve x^2 and y^2, which indicates that the solutions may involve quadratic relationships. Understanding how to manipulate and solve quadratic equations is essential for finding the values of x and y in this context.