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Chapter 1: Problem 25

Find the value for \((a) x=2\) and \(y=1\) and \((b) x=1\) and \(y=5\). \(3(x+2 y)\)

Short Answer

Expert verified
For (a), the value is 12. For (b), the value is 33.

Step by step solution

01

Understand the Expression

The given expression is \(3(x + 2y)\). This means we need to first add \(2y\) to \(x\) and then multiply the result by 3.
02

Step 2a: Substitute Values for (a) x=2 and y=1

In part (a), we are given \(x=2\) and \(y=1\). Substitute these values into the expression: \[ 3(2 + 2 \times 1) \].
03

Step 2b: Simplify the Expression for (a)

First, calculate the inside of the parentheses: \[ 2 + 2 \times 1 = 2 + 2 = 4 \]. Then, multiply this result by 3: \[ 3 \times 4 = 12 \].
04

Step 3a: Substitute Values for (b) x=1 and y=5

In part (b), we are given \(x=1\) and \(y=5\). Substitute these values into the expression: \[ 3(1 + 2 \times 5) \].
05

Step 3b: Simplify the Expression for (b)

First, calculate the inside of the parentheses: \[ 1 + 2 \times 5 = 1 + 10 = 11 \]. Then, multiply this result by 3: \[ 3 \times 11 = 33 \].

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Key Concepts

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

Substitution
Substitution is an essential part of solving algebraic expressions. It involves replacing variables with their given values. When we substitute values into an expression, we find particular solutions. For example, in our exercise: we were given specific values for x and y. In part (a), we replaced x with 2 and y with 1. For part (b), we substituted x with 1 and y with 5.
To substitute correctly:
  • Identify the variables in the expression.
  • Replace each variable with the given number.
  • Ensure you keep the rest of the expression unchanged.
By substituting accurately in our example, we could further simplify and solve the expression.
Simplification
Simplification makes an expression easier to work with by reducing it to its simplest form. It often follows after you have performed substitution. For instance, in our step-by-step solution, after substituting the values, we simplified the expression inside the parentheses first.
Simplifications usually follow these steps:
  • Perform any operations inside parentheses.
  • Carry out multiplication or division from left to right.
  • Finish with any addition or subtraction.
In part (a), after substituting, we had \(2 + 2 \times 1\). We calculated inside the parentheses first to simplify to \(4\) before multiplying by 3. Simplification made it simple to find the final value.
Multiplication
Multiplication is one of the core arithmetic operations used in algebraic expressions. After we performed substitution and simplification, multiplication was the next step. In our expression, we had to multiply after producing the simplified result inside the parentheses.
To multiply correctly:
  • Ensure you have simplified the expressions inside any parentheses.
  • Multiply the numbers directly following arithmetic rules.
  • Keep track of sign changes, especially if dealing with negative numbers.
For part (a), we multiplied 3 by the simplified result: 3 multiplied by 4 gave us 12. Multiplication, when done right, ensures you get your final correct answer.

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