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Chapter 1: Problem 61

Decide whether the given number is a solution of the equation. \(3 r^{2}-2=46 ; 4\)

Short Answer

Expert verified
Yes, \(4\) is a solution.

Step by step solution

01

- Understand the Equation

The given equation is: \(3r^{2} - 2 = 46\). We need to check if \(r = 4\) satisfies this equation.
02

- Substitute the Value of r

Substitute \(r = 4\) into the equation: \(3(4)^{2} - 2 = 46\).
03

- Calculate the Exponent

Calculate \((4)^{2}\): \((4)^{2} = 16\).
04

- Multiply by the Coefficient

Multiply 16 by 3: \(3 \times 16 = 48\).
05

- Simplify the Expression

Perform the subtraction: \(48 - 2 = 46\).
06

- Compare Both Sides

Compare the simplified left-hand side (46) with the right-hand side (46). Since they are equal, \(r = 4\) is a solution to the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a crucial approach when solving equations, especially quadratic equations. Let's break it down step-by-step:

What is Substitution?
Substitution involves replacing a variable with a given value to simplify and solve the equation. In our example, we are given the equation: \(3r^{2}-2=46\). We aim to see if \(r=4\) is a solution.

Step-by-Step Process
First, substitute the value of \(r\) into the equation. This means every instance of \(r\) in \(3r^{2}-2=46\) is replaced with 4, resulting in \(3(4)^{2} - 2 = 46\).

This method helps in verifying if our given value satisfies the equation by simplifying it. It's an essential skill for solving algebraic problems correctly.
Exponents
Understanding exponents is fundamental in solving equations involving powers. Let's dive into the concept and our problem.

What are Exponents?
Exponents are used to express repeated multiplication of a number by itself. For example, \(4^{2}\) means multiplying 4 by itself: \(4 \times 4 = 16\).

Applying Exponents
In our example, we need to calculate \(4^{2}\) as part of solving \(3r^{2}-2=46\). Substituting \(r = 4\) into the equation, we get \(3(4)^{2} - 2 = 46\). Here, \(4^{2} = 16\), simplifying the expression further.

Exponents simplify complex expressions and are used widely in algebra to manage large numbers.
Equation Verification
Once you've worked through the substitution and exponent calculations, the final step is equation verification.

Why Verify Equations?
Verification ensures that the solution holds true for the entire equation. Our goal is to make sure both sides of the equation equal each other after substitution.

Verification in Action
In the problem \(3r^{2}-2=46\) with \(r=4\), we've simplified the expression to \(3(16)-2=46\). This further simplifies to \(48-2=46\) which is true since 46 equals 46.

This step reassures us that our calculated value of \(r\) is indeed correct and confirms the solution.

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Most popular questions from this chapter

Write a numerical expression for each phrase and simplify. Three subtracted from the product of 4.2 and \(-8.5\)

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ 6 x-5 y+4 a $$

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ \left(\frac{1}{3} x-\frac{4}{5} y\right)\left(-\frac{1}{5} a\right) $$

Perform each indicated operation. $$ 4(-8)+|4-15| $$

To find the average (mean) of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. $$ -15,29,8,-6 $$
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