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Chapter 0: Problem 45

Write each statement using symbols. The product of 3 and \(y\) is the sum of 1 and 2 .

Short Answer

Expert verified
3y = 3

Step by step solution

01

- Identify the Variables

The problem involves the variable 'y'. Note that '3' and '2' are constants.
02

- Translate Words to Mathematical Operations

The word 'product' indicates multiplication. So, 'the product of 3 and y' can be written as '3y'. The word 'sum' indicates addition. 'The sum of 1 and 2' can be written as '1 + 2'.
03

- Write the Entire Equation

Combine the mathematical expressions from Step 2 into one equation. 'The product of 3 and y is the sum of 1 and 2' translates to the equation: 3y = 1 + 2.
04

- Simplify the Equation

Simplify the right side of the equation: 1 + 2 = 3 Therefore, the equation simplifies to: 3y = 3.

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Key Concepts

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

Translating Words to Equations
Translating words into mathematical equations is a crucial skill in algebra. This process involves converting a verbal statement into its numerical counterpart.

Consider the statement: 'The product of 3 and y is the sum of 1 and 2.' Here, 'product' means multiplication, and 'sum' means addition.

First, identify key terms:
  • 'The product of 3 and y' means \(3y\).
  • 'The sum of 1 and 2' means \(1 + 2\).
Combining these, we get: \(3y = 1 + 2\). With practice, recognizing these patterns becomes second nature, allowing you to easily translate complex verbal statements into equations.
Simplifying Equations
Once you've translated a statement into an equation, the next step is simplification.

From our earlier example, the equation \(3y = 1 + 2\) needs to be simplified. Simplifying involves performing any arithmetic operations to combine like terms and make the equation easier to work with.

In this case, we simplify the right side of the equation. Adding 1 and 2 gives us 3. So, \(3y = 1 + 2\) simplifies to \(3y = 3\).

Simplification is important because it helps reduce an equation to its simplest form, making it easier to solve.
Identifying Variables
Identifying variables is the first step in solving any algebraic problem. A variable represents an unknown quantity that you need to find.

In the given statement, 'The product of 3 and y is the sum of 1 and 2,' the variable is 'y.' Variables are usually letters like x, y, z, but they can be any symbol.

Recognizing which symbol is the variable lets you set up your equation correctly. In more complex problems, there may be multiple variables. But in this example, it's straightforward with just one variable, y.

To summarize, always look for the unknowns in the problem and assign appropriate variables to them. This makes the rest of the problem much easier to handle.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$6(6 x+1)^{1 / 3}(4 x-3)^{3 / 2}+6(6 x+1)^{4 / 3}(4 x-3)^{1 / 2} \quad x \geq \frac{3}{4}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x}-2}{x-4} x \neq 4$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{6}-\sqrt{15}}{\sqrt{15}}$$

Simplify each expression. $$\left(\frac{8}{9}\right)^{-3 / 2}$$
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