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Mobile App Chapter 3: Problem 72
If \((1 / a)+(1 / b)=2\) and \((2 / a)+(3 / b)=5, b\) is: (A) 3 (B) 4 (C) 2 (D) 1 (E) \(1 / 2\)
Short Answer
Expert verified
b = 1 (Option D)
Step by step solution
01
Write down the given equations
We start with the given equations: 1. \(\frac{1}{a} + \frac{1}{b} = 2\)2. \(\frac{2}{a} + \frac{3}{b} = 5\)
02
Substitute the first equation
Rearrange the first equation to express \(\frac{1}{a}\) in terms of \(\frac{1}{b}\): \[\frac{1}{a} = 2 - \frac{1}{b}\]
03
Substitute into the second equation
Replace \(\frac{1}{a}\) in the second equation with the expression from Step 2: \[(2(2 - \frac{1}{b}) + \frac{3}{b} = 5)\]
04
Simplify the equation
Expand and simplify the equation: \[4 - \frac{2}{b} + \frac{3}{b} = 5\]Combine the fraction terms: \[4 + \frac{1}{b} = 5\]
05
Solve for \(\frac{1}{b}\)
Isolate \(\frac{1}{b}\) on one side: \[\frac{1}{b} = 5 - 4 = 1\]
06
Solve for \(b\)
Take the reciprocal of both sides: \[b = 1\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
equation solving
Equation solving is the process of finding the values of variables that satisfy a given equation or set of equations. In our exercise, we are given two equations involving the variables a and b, and we need to find the value of b. This involves manipulating the equations using basic algebra to isolate and solve for the variables. Understanding how to solve equations is essential because it helps you work through problems methodically and reach the correct answer. Always start by writing down the given equations clearly. This helps in visualizing your path to the solution.
reciprocal functions
Reciprocal functions are functions of the form \(\frac{1}{x}\). They are essential in this exercise as both given equations involve terms like \(\frac{1}{a}\) and \(\frac{1}{b}\). Reciprocal functions are inverted functions, meaning if you have \(x\), its reciprocal is \(\frac{1}{x}\). Understanding how reciprocal functions work is crucial when solving equations that involve dividing by a variable. In the exercise, we manipulated these reciprocal expressions to form a solvable equation. For example, when we stated \(\frac{1}{a} = 2 - \frac{1}{b}\), we were working directly with the reciprocal relations.
substitution method
The substitution method is a technique used in solving systems of equations where one equation is substituted into another. This allows you to solve for one variable at a time. In our example, we started by isolating \(\frac{1}{a}\) from the first equation and then substituting this expression into the second equation. This step-by-step approach helps simplify the system, making it easier to find the solution. To use substitution effectively:
- Solve one of the equations for one variable.
- Substitute this expression into the other equation.
- Simplify and solve the resulting equation.
math problem steps
Following a structured approach to solving math problems ensures you reach the solution methodically. Let's break down our exercise step by step:
- Step 1: Write down the given equations. This gives you a clear starting point.
- Step 2: Solve one of the equations for a variable. In our case, \(\frac{1}{a}\).
- Step 3: Substitute the solved variable into the other equation. This reduces the number of variables.
- Step 4: Simplify the substituted equation. Combine like terms and reduce fractions to make the equation easier to handle.
- Step 5: Solve for the variable. Isolate the variable on one side to find its value.
- Step 6: Verify your solution by substituting it back into the original equations. This ensures that your solution is correct.
