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Chapter 2: Problem 19

If \(a+b+c=12\), find the value of $$(a+5)+(b-3)+(c+8)$$

Short Answer

Expert verified
Question: Given that \(a+b+c=12\), find the value of \((a+5)+(b-3)+(c+8)\). Answer: 22

Step by step solution

01

Rewrite the expression

We can rewrite \((a+5)+(b-3)+(c+8)\) as \(a+b+c+5-3+8\).
02

Simplify the expression

Simplify the expression by combining the constants: $$a+b+c+5-3+8 = a+b+c+10$$
03

Substitute the given relationship

Since we know that \(a+b+c=12\), we can substitute this into the expression: $$12+10$$
04

Calculate the final value

Now, simply add the numbers: $$12+10 = 22$$ Therefore, the value of \((a+5)+(b-3)+(c+8)\) is 22.

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Key Concepts

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

Simplifying Expressions
When working with algebraic expressions, simplifying them can make calculations easier and more intuitive.
For example, in the given expression \((a+5)+(b-3)+(c+8)\), we want to combine like terms to make it more straightforward.
  • Start by identifying similar components in the expression: the variables \(a\), \(b\), and \(c\), and the constants 5, -3, and 8.
  • Combine the constants: 5, -3, and 8. Add and subtract these numbers: \(5 - 3 + 8\), which simplifies to 10.
Once you have combined the constants, you can rewrite the expression as \(a+b+c+10\).
This new form is simpler and easier to handle.
By reducing unnecessary complexity, simplification makes solving algebra problems a more manageable task.
Substitution Method
The substitution method is a vital concept in algebra, especially when dealing with expressions and equations.
It involves replacing a variable with a given value or expression to simplify calculations. In our original problem, we are given that \(a+b+c=12\).
We can use the substitution method to make our expression easier to solve by replacing \(a+b+c\) with 12.
  • Write down the simplified expression: \(a+b+c+10\).
  • Substitute \(a+b+c\) with 12, based on the given equation: \(12+10\).
This step directly utilizes known relationships among variables to simplify and solve expressions.
The substitution method is efficient because it leverages existing information to lead directly to the solution.
Linear Equations
Linear equations are a foundational element of algebra that involve variables with an exponent of 1, such as the equation \(a+b+c = 12\).
These equations can be solved using simple algebraic techniques. The given problem does not require solving the linear equation for individual values of \(a\), \(b\), or \(c\), but instead uses the equation to simplify another expression.
  • Understand that \(a+b+c=12\) forms a key component for the transformation of the expression we initially started with.
  • Note that all variables are linear, meaning they are not raised to any power other than 1, which keeps the equation straightforward.
Grasping linear equations prepares you for more advanced algebra topics, as it teaches the handling of basic equations and their relationships.
They serve as a stepping stone to understanding more complex expressions and systems of equations.

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Most popular questions from this chapter

Rewrite the expression by taking out the common factors. \(5 x+100\)

If \(x y z=100,\) find the value of \((3 x)(2 y)(5 z)\).

Are the two expressions equivalent? \(5 x+100\) and \(5(x+500)\)

Write each expression as a sum, difference, or product of two or more algebraic fractions. There is more than one correct answer. Assume all variables are positive. $$ \frac{8-5 x}{-2} $$

In Exercises \(9-19\), rewrite the expression by taking out the common factors. \(2 a x-3 b x\)
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