Problem 37 - Intermediate Algebra

Question

For Exercises 37 through \(42,\) first divide the equation through by the coefficient of \(\left.x^{2} \text { (or } y^{2}\right)\) $$3 x^{2}+3 y^{2}=75$$

Step-by-Step Solution

Answer

The simplified equation is \(x^2 + y^2 = 25\).

Step 1: Identify the Coefficient of \(x^2\) and \(y^2\)

In the given equation, \(3x^2 + 3y^2 = 75\), the coefficient of both \(x^2\) and \(y^2\) is \(3\).

Step 2: Divide Each Term by the Coefficient

Divide each term in the equation by \(3\), the coefficient of \(x^2\) and \(y^2\), to simplify the equation. This results in:\[\frac{3x^2}{3} + \frac{3y^2}{3} = \frac{75}{3}\]

Step 3: Simplify the Equation

Simplify the equation by performing the division for each term:\[x^2 + y^2 = 25\]

Key Concepts

Understanding Coefficients

Coefficients are numbers or constants placed before variables in algebraic expressions or equations. They play a significant role in determining the specific characteristics of the equation. For example, in the expression \(3x^2 + 3y^2 = 75\), the number '3' is considered the coefficient of both \(x^2\) and \(y^2\). This tells us by how much these terms are multiplied.

• They are crucial in simplifying equations and solving algebraic expressions.
• Understanding coefficients is essential, especially in resolving quadratic equations, because they affect how the equation behaves.

In general, coefficients can be integer, fractional, or irrational numbers. Identifying the coefficients of an equation is a key step in any solution process.

Simplifying Equations

Simplifying equations is about making the equation easier to work with while maintaining its fundamental qualities. It involves breaking down complex terms and expressions into simpler forms.
One of the first steps in simplifying an equation is addressing any coefficients that might be present. As seen in our example \(3x^2 + 3y^2 =75\), each term carries the same coefficient.

• To simplify, divide each term by its common coefficient – in this case, we divide by \(3\).
• This transforms the left-hand side of the equation to \(x^2 + y^2\), and the right-hand side to \(25\).

So, the simplified equation becomes \(x^2 + y^2 = 25\), which is easier to interpret and solve.

Step-by-Step Solutions

Approaching quadratic equations with a step-by-step mindset can significantly enhance comprehension and problem-solving skills. When presented with an equation like \(3x^2 + 3y^2 = 75\), it's essential to methodically tackle each segment of the problem.

• Start by identifying the coefficients, as these can influence both the structure and the method needed for simplification.
• Continue by evenly distributing any necessary operations across all terms; for instance, dividing by a common factor.
• Simplify each part of the equation individually to ensure accuracy and clarity.

This structured approach builds foundational skills that are crucial as students encounter more complex mathematical problems. Remember, practice makes perfect, so apply these steps regularly for better results.