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Chapter 3: Problem 2

Solve the equations. \(\frac{7}{10}=\frac{7}{5} k\)

Short Answer

Expert verified
k = \frac{1}{2}

Step by step solution

01

Understand the Equation

The given equation is \( \frac{7}{10} = \frac{7}{5} k \). We aim to find the value of \( k \) that satisfies this equation.
02

Isolate the Variable

To isolate \( k \), divide both sides of the equation by \( \frac{7}{5} \). This gives us: \[ k = \frac{ \frac{7}{10} }{ \frac{7}{5} } \]
03

Simplify the Fraction

To simplify \( \frac{ \frac{7}{10} }{ \frac{7}{5} } \), multiply the numerator by the reciprocal of the denominator: \[ k = \frac{7}{10} \times \frac{5}{7} \]
04

Perform the Multiplication

Multiply the fractions: \[ k = \frac{7 \cdot 5}{10 \cdot 7} = \frac{35}{70} \]
05

Simplify the Result

Simplify the fraction \( \frac{35}{70} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 35: \[ k = \frac{35 \div 35}{70 \div 35} = \frac{1}{2} \]

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Key Concepts

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

Algebraic Equations
Algebraic equations consist of variables and constants combined using arithmetic operations. In the given exercise, we have the equation \(\frac{7}{10} = \frac{7}{5} k\). Our goal is to find the value of the variable \(k\) that makes the equation true. Understanding algebraic equations is crucial because they represent relationships between quantities. These relationships help us solve for unknowns by applying mathematical operations while maintaining the balance of the equation. Initially, equating both sides helps us direct our approach to solve for \(k\).
Fractions
Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). In the exercise, we see fractions like \(\frac{7}{10}\) and \(\frac{7}{5}\). When solving equations with fractions, it鈥檚 essential to understand how to manipulate them, such as multiplying, dividing, and simplifying. For instance, to simplify \(\frac{\frac{7}{10}}{\frac{7}{5}}\), you multiply by the reciprocal of the divisor. This is a technique used frequently in algebra to make equations more manageable.
Isolation of Variables
Isolation of variables means rearranging an equation to put the variable you are solving for on one side and everything else on the other. In our exercise, to isolate \(k\), we divide both sides by \(\frac{7}{5}\). This step simplifies the equation to represent \(k\) as the subject. Isolating variables is fundamental in algebra because it allows us to express the unknown variable explicitly and solve the equation step-by-step. For example, starting from \(\frac{7}{10} = \frac{7}{5} k\), dividing both sides by \(\frac{7}{5}\) gives us \(k = \frac{\frac{7}{10}}{\frac{7}{5}}\).
Simplification
Simplifying an equation makes it easier to solve. In the solution, we simplified \( \frac{\frac{7}{10}}{\frac{7}{5}}\) to \(\frac{7}{10} \times \frac{5}{7}\). This step includes finding the reciprocal of \(\frac{7}{5}\) and multiplying it with \(\frac{7}{10}\). Simplifications often involve reducing fractions to their lowest terms. In the last part of our solution, we simplify \(\frac{35}{70}\) to \(\frac{1}{2}\) by dividing the numerator and denominator by their greatest common divisor, which is 35. Understanding how to simplify ensures that our final answer is in its simplest form, making it easier to understand and verify.

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Most popular questions from this chapter

Which formula(s) can represent a variation model? a. \(y=k x y z\) b. \(y=k x+y z\) c. \(y=\frac{k x}{y z}\) d. \(y=k x-y z\)

Determine if the statement is true or false. a. Use the intermediate value theorem to show that \(f(x)=\) \(2 x^{2}-7 x+4\) has a real zero on the interval [2,3] . b. Find the zeros.

Sometimes it is necessary to use a "friendly" viewing window on a graphing calculator to see the key features of a graph. For example, for a calculator screen that is 96 pixels wide and 64 pixels high, the "decimal viewing window" defined by [-4.7,4.7,1] by [-3.1,3.1,1] creates a scaling where each pixel represents 0.1 unit. The window [-9.4,9.4,1] by [-6.2,6.2,1] defines each pixel as 0.2 unit, and so on. Exercises \(112-113\) compare the use of the standard viewing window to a "friendly" viewing window. a. Identify any vertical asymptotes of the function defined by \(f(x)=\frac{x^{2}+3 x+2}{x+1}\). b. Compare the graph of \(f(x)=\frac{x^{2}+3 x+2}{x+1}\) on the standard viewing window [-10,10,1] by [-10,10,1 and on the window [-4.7,4.7,1] by [-3.1,3.1,1] . Which graph shows the behavior at \(x=-1\) more completely?

Write a statement in words that describes the variation model given. Use \(k\) as the constant of variation. \(P=\frac{k v^{2}}{t}\)

Sometimes it is necessary to use a "friendly" viewing window on a graphing calculator to see the key features of a graph. For example, for a calculator screen that is 96 pixels wide and 64 pixels high, the "decimal viewing window" defined by [-4.7,4.7,1] by [-3.1,3.1,1] creates a scaling where each pixel represents 0.1 unit. The window [-9.4,9.4,1] by [-6.2,6.2,1] defines each pixel as 0.2 unit, and so on. Exercises \(112-113\) compare the use of the standard viewing window to a "friendly" viewing window. a. Identify any vertical asymptotes of the function defined by \(f(x)=\frac{x^{2}-5 x+4}{x-4}\) b. Compare the graph of \(f(x)=\frac{x^{2}-5 x+4}{x-4}\) on the standard viewing window [-10,10,1] by [-10,10,1] and on the window [-9.4,9.4,1] by [-6.2,6.2,1] . Which graph shows the behavior at \(x=4\) more completely?
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