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Chapter 7: Problem 11

\(x=12, y=10,\) and \(z=24 .\) Write each ratio in simplest form. $$\frac{y+z}{x-y}$$

Short Answer

Expert verified
The simplified form of the ratio \(\frac{y+z}{x-y}\), where \(x = 12\), \(y = 10\), and \(z = 24\), is 17.

Step by step solution

01

Substitute values

First, substitute the given values \(x = 12\), \(y = 10\), and \(z = 24\) into the ratio\(\frac{y+z}{x-y}\) :\[\frac{10+24}{12-10}\]This will yield: \[\frac{34}{2}\]
02

Simplify the fraction

Second, you will simplify \(\frac{34}{2}\) by dividing 34 by 2. The result will therefore be 17

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a fundamental concept in mathematics. It involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This is important because simplified fractions are easier to work with and understand.

To simplify a fraction, follow these general steps:
  • Find the greatest common factor (GCF) of the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCF.
  • Write the resulting fraction, which should now be in its simplest form.
For example, when we have the fraction \(\frac{34}{2}\), we can see that both 34 and 2 can be divided by 2 (since 2 is a factor of both numbers). By dividing the numerator and the denominator by 2, we simplify this to \(\frac{17}{1}\) or simply 17.
Substituting Values
Substituting values is a straightforward yet crucial technique in algebra and mathematics. It involves replacing the variables in an expression or equation with given numerical values to simplify the problem or find a specific solution. This process helps to concretely evaluate expressions.

Take these steps when substituting values:
  • Identify the variables in the expression or equation that have given values.
  • Carefully replace each variable with its corresponding numerical value.
  • Simplify the expression by performing basic arithmetic operations.
For instance, with the expression \(\frac{y+z}{x-y}\), given \(x = 12\), \(y = 10\), and \(z = 24\), substituting the values gives \(\frac{10 + 24}{12 - 10}\). This substitution converts the algebraic fraction into a simple numerical fraction \(\frac{34}{2}\), which we can then simplify further.
Basic Algebra
Basic algebra includes the use of variables to represent numbers in equations and expressions, making it possible to solve a wide range of problems. This branch of mathematics forms the foundation for many advanced topics.

Some key elements of basic algebra include:
  • Variables: Symbols or letters that stand in for unknown or changeable values.
  • Expressions: Combinations of variables, numbers, and operations (like addition or multiplication).
  • Equations: Statements that show two expressions are equal, often involving an unknown to solve for.
When dealing with expressions such as \(\frac{y+z}{x-y}\), understanding basic algebra helps us apply operations correctly. It guides us through substituting values, simplifying expressions, and interpreting the meaning of the variable-based expressions correctly. By practicing algebra, students develop not only calculation skills but also logical problem-solving abilities.

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Most popular questions from this chapter

Complete each statement. If \(\frac{a}{4}=\frac{b}{7},\) then \(\frac{a}{b}=\frac{?}{?}\)

The lengths of the sides of \(\triangle A B C\) are \(B C=12 . C A=13\) and \(A B=14 .\) If \(M\) is the midpoint of \(\overline{C A}\). and \(P\) is the point where \(\overline{C A}\) is cut by the bisector of \(\angle B\). find \(M P\).

Find the values of \(x\) and \(y\). $$\begin{aligned} &\frac{x-3}{4}=\frac{y+2}{2}\\\ &\frac{x+y-1}{6}=\frac{x-y+1}{5} \end{aligned}$$

Prove that there cannot be a triangle in which the trisectors of an angle also trisect the opposite side.

Write the algebraic ratio in simplest form. $$\frac{3(x+4)}{a(x+4)}$$
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