Problem 55 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{15 x^{2}+10}{y} \text { when } x=-3 \text { and } y=\frac{2}{3}$$

Step-by-Step Solution

Answer

The evaluated expression when \(x=-3\) and \(y=\frac{2}{3}\) equals 217.5

Step 1: Substitute the given values

The first step is to substitute x with -3 and y with \(2/3\) in the expression \(\frac{15 x^{2}+10}{y}\). Doing so, we get: \(\frac{15*(-3)^{2}+10}{2/3}\)

Step 2: Simplify Above Expression

Now simplify the equation. Solve for \((-3)^{2}\) which gives 9, then multiply 15 by 9 and add 10. This gives: \(\frac{15*9+10}{2/3} = \frac{145}{2/3}\)

Step 3: Simplification

Now divide 145 by \(2/3\). Since dividing by a fraction is the same as multiplying by its reciprocal, it gives: \(145 * \frac{3}{2} = 217.5\)

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to simplify and solve expressions or equations. This method involves replacing variables with their given values to evaluate the expression. In the problem at hand, we are given specific values for the variables \(x\) and \(y\).
By substituting \(x = -3\) and \(y = \frac{2}{3}\) into the expression \(\frac{15x^2 + 10}{y}\), we replace each variable with its respective value. This transforms the algebraic expression into a numerical one, which is easier to handle and solve.

• Start with the original expression: \(\frac{15x^2 + 10}{y}\).
• Substitute \(x = -3\) to get: \(15(-3)^2 + 10\).
• Substitute \(y = \frac{2}{3}\) for the entire denominator.

After substitution, operations can be carried out to solve the expression and find the numerical result. This highlights the importance of precision when replacing variables to ensure accuracy in the final result.

Rational Expressions

Rational expressions are fractions that involve polynomials in the numerator and denominator. In this exercise, the expression \(\frac{15x^2 + 10}{y}\) is a rational expression, as it includes a polynomial \(15x^2 + 10\) in the numerator and a variable \(y\) in the denominator.

• Rational expressions require careful operations, especially when substituting values.
• When substituting \(y\) with \(\frac{2}{3}\), the entire denominator becomes a simple fraction.

Performing operations with fractions needs a solid understanding of multiplying and dividing polynomials, particularly since the division by a fraction is equivalent to multiplying by its reciprocal. After substitution, the expression becomes \(\frac{145}{\frac{2}{3}}\), which is then simplified by multiplying \(145\) by \(\frac{3}{2}\). This approach simplifies the rational expression to achieve the final result.

Arithmetic Operations

Arithmetic operations are the basic mathematical tasks such as addition, subtraction, multiplication, and division used to solve equations and expressions. In evaluating the expression \(\frac{15x^2 + 10}{y}\), several arithmetic operations are performed.
First, calculate \((-3)^2\) to get 9, then multiply by 15 to give 135, and add 10 results in a total of 145. This use of addition and multiplication follows the correct order of operations (PEMDAS/BODMAS).
Next, division is carried out by converting division by a fraction into multiplication by its reciprocal. This means that \(\frac{145}{\frac{2}{3}}\) becomes \(145 \times \frac{3}{2}\), resulting in the final answer of 217.5. Understanding these steps demonstrates the power of arithmetic operations in achieving simplification and clarity in mathematical problems.

• Square operation first (exponents).
• Then multiplication, followed by addition.
• Finally, simplify by division using multiplication of reciprocals.

Mastery of these operations ensures efficiency in handling similar algebraic expressions.