Problem 38 - Intermediate Algebra

Question

For Exercises 37 through \(42,\) first divide the equation through by the coefficient of \(\left.x^{2} \text { (or } y^{2}\right)\) $$2 x^{2}+2 y^{2}=18$$

Step-by-Step Solution

Answer

The simplified equation is \(x^2 + y^2 = 9\).

Step 1: Identify the Coefficient

First, identify the coefficient of the term with the variable squared. In this equation, the terms are \(2x^2\) and \(2y^2\), so the coefficient of \(x^2\) (or \(y^2\)) is 2.

Step 2: Divide the Equation by the Coefficient

To simplify the equation, divide both sides by the coefficient of \(x^2\) (or \(y^2\)). This means we divide every term in the equation by 2:\[\frac{2x^2}{2} + \frac{2y^2}{2} = \frac{18}{2}\] which simplifies to \(x^2 + y^2 = 9\).

Key Concepts

Quadratic Equations

Quadratic equations are a key concept in intermediate algebra. They play a significant role in various fields of mathematics and applications such as physics and engineering. These types of equations are typically written in the standard form: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The task is to find the values of \(x\) that make the equation true.

A unique feature of quadratic equations is their graphical representation. On a graph, a quadratic equation forms a parabola. The direction in which the parabola opens depends on the coefficient \(a\). If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards. Understanding this graph provides insights into the behavior of quadratic functions.

To solve quadratic equations, several methods can be employed such as factoring, using the quadratic formula, or completing the square. In the context of your exercise, you've worked with an equation involving squared terms \(x^2\) and \(y^2\). By organizing and simplifying such terms, we often transform these equations into more manageable forms.

Coefficients

In algebra, coefficients are the numeric factors in terms involving variables. They help us understand how the variable impacts the expression. In the equation \(2x^2 + 2y^2 = 18\), for example, the number 2 is the coefficient for both \(x^2\) and \(y^2\). This tells us that each term was scaled by a factor of 2.

Recognizing and manipulating coefficients is a fundamental skill in algebra. It aids in simplification, comparison, and graphing of equations. To simplify, dividing each term of an equation by its common coefficient can yield a simpler equivalent equation which is easier to solve.

When you identify coefficients, it's crucial also to note their role in multi-variable equations. In such cases, they determine the geometric properties of the graph. For example, equal coefficients on \(x^2\) and \(y^2\) often imply symmetry in the equation, such as in circles or ellipses.

Simplifying Equations

Simplifying equations is an important skill in solving algebraic problems. It involves rewriting an equation in a form that is easier to work with or understand, while maintaining its equivalence. This process often includes combining like terms, reducing fractions, or factoring expressions.

In the provided exercise, simplifying was achieved by dividing every term by the coefficient 2. This reduced the equation \(2x^2 + 2y^2 = 18\) into \(x^2 + y^2 = 9\). This form is much more recognizable and can be interpreted as the equation of a circle with a radius of \(3\) (since \(\sqrt{9} = 3\)) centered at the origin.

The goal of simplifying is not just to make calculations more manageable. It also helps in revealing the nature or the solution of the problem. Always aim for concise expressions while ensuring all variables and their interactions are maintained accurately.