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Chapter 2: Problem 20

If \(x+y+z=25\), find the value of $$(y-10)+(z+8)+(x-5)$$

Short Answer

Expert verified
Answer: The value of \((y-10)+(z+8)+(x-5)\) is 18.

Step by step solution

01

Rewrite the expression

We need to rewrite the expression \((y-10)+(z+8)+(x-5)\) in a way that we can use the given equation \(x+y+z=25\). Let's add these terms separately to understand their relationship better. \begin{align} (y-10) + (z+8) + (x-5) &= y - 10 + z + 8 + x - 5\\ &= x + y + z - 7 \end{align} Now, we have the expression in terms of \(x+y+z\).
02

Substitute the given value

We are given that \(x+y+z=25\). Substitute this value into the rewritten expression from Step 1: \begin{align} x + y + z - 7 &= 25 - 7 \\ &= 18 \end{align}
03

State the final answer

Thus, after simplifying the given expression and substituting the given value, we find that \((y-10)+(z+8)+(x-5) = 18\).

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Key Concepts

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

Equation Solving
Solving equations is a critical skill in algebra that involves finding the values of variables that make an equation true. In this exercise, we started with the simple equation given:
  • Step 1: We needed to use the equation \(x+y+z=25\). This serves as the foundation for manipulating other parts of the problem.
  • Step 2: We use mathematical operations to deduce other unknowns from this primary equation.
Equation solving often involves rearranging the equation to isolate the unknown. This means performing operations like adding, subtracting, multiplying, or dividing. Always remember to perform any operation equally on both sides to maintain equality. The beauty of algebra lies in these transformations. They allow us to turn complex problems into simple calculations.
Variable Substitution
Variable substitution is a process that involves replacing a variable with a known value or another expression. Here, we had to substitute the expression \(x+y+z\) with the number 25. This substitution took place after rewriting the original expression
  • This helps to simplify problems by leveraging known quantities to replace variables.
  • In this exercise, once we knew that \(x+y+z=25\), we could confidently swap the sum in our new expression with 25.
By substituting, we transform the problem from a seemingly more complex equation with multiple variables into a simple arithmetic problem. Once variables are replaced by numbers, it becomes straightforward to solve resulting expressions. This illustrates how substitution can be a powerful tool in breaking down problems.
Simplifying Expressions
Simplifying an expression means making it as straightforward and concise as possible. This involves combining like terms or using known mathematical identities. The original expression was
  • \((y-10)+(z+8)+(x-5)\) which was rewritten as \(x+y+z-7\).
  • Simplification often unveils easier pathways to complete a solution.
When we apply rules of arithmetic, such as combining terms and using distributive properties, complex expressions become less intimidating. Here, by adding and subtracting constants, we ended up with a simpler form: \(x+y+z-7\). Simplification makes handling expressions more efficient and helps in quickly evaluating or understanding their essence.

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Most popular questions from this chapter

Simplify each expression. Assume any factors you cancel are not zero. $$ 2-\frac{2}{2+\frac{2}{2+2}} $$

Are the two expressions equivalent? \(3 x^{2}+6 x+3\) and \(3\left(x^{2}+2 x\right)\)

Factor each expression completely. $$ (t+1)^{3}-25(t+1) $$

Factor the expressions in exercises. $$ x^{2}-169 $$

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