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Chapter 3: Problem 4

Solve each equation for \(b\) $$-4=\frac{3}{2}(-5)+b$$

Short Answer

Expert verified
The solution to the equation is b = 3.5.

Step by step solution

01

Understand and Simplify

First, let's understand the equation that is given as -4 = \(\frac{3}{2}(-5) \) + b. This can be simplified further. Simplify \( \frac{3}{2}(-5) \). Multiply 3/2 and -5. The multiplication comes first in order of operations, so this can be done first.
02

Calculation

Calculate \( \frac{3}{2}* (-5) \) which equals to -7.5. So the equation becomes -4 = -7.5 + b.
03

Isolate the variable

As the purpose is to isolate b, add 7.5 to both sides of the equation to get rid of -7.5 on the right side. So, you get 3.5 = b.

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Key Concepts

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

Understanding the Order of Operations
When solving linear equations like \(-4 = \frac{3}{2}(-5) + b\), it's crucial to apply the order of operations correctly. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), can help guide this process. The order of operations tells us how to simplify expressions properly and avoid common mistakes.
  • First, look inside any parentheses and perform operations like multiplication. In our case, this involves multiplying \(\frac{3}{2}\) by \(-5\).
  • By simplifying \(\frac{3}{2} \times (-5)\), you prioritize multiplication, resulting in \(-7.5\).
  • Once this part of the expression is simplified, substitute this back into the equation, leading to a simpler equation \(-4 = -7.5 + b\).
Understanding the order in which calculations are performed allows for more accurate and efficient problem-solving.
Simplifying Expressions
Simplifying expressions makes solving equations easier and more straightforward. For the given equation \(-4 = \frac{3}{2}(-5) + b\), you began by simplifying the expression \(\frac{3}{2} (-5)\). This step requires attention to multiplication within the expression.
  • Calculate \(\frac{3}{2} \times (-5)\) to simplify the expression. This results in \(-7.5\).
  • This turns the original equation into \(-4 = -7.5 + b\).
Simplification reduces complex expressions into simpler terms, removing barriers to finding the solution. This process ensures you have a clear path forward to isolate and solve for your variable.
Isolating Variables
The ultimate goal of solving a linear equation is to isolate the variable you are solving for. In the exercise, your task is to solve for \(b\). Isolating the variable makes it possible to identify its value clearly and accurately.
  • From the simplified equation \(-4 = -7.5 + b\), you must eliminate \(-7.5\) from the right side.
  • By adding \(7.5\) to both sides of the equation, you effectively cancel \(-7.5\) and isolate \(b\) on the right side. The equation then becomes \(3.5 = b\).
Isolating \(b\) ensures that the solution is directly and neatly obtained. Each step should carefully balance the equation and ensure that operations performed keep the equation equivalent. This focus on balance and equality allows for exact results, providing clarity in the solution.

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Most popular questions from this chapter

From geometry, we know that two points determine a line. Explain why it is good practice when graphing linear equations to find and plot three points instead of just two.

Research A psychology major found that the time \(t\) in seconds that it took a white rat to complete a maze was related to the number of trials \(n\) the rat had been given by the equation \(t=25-0.25 n\) a. Complete the table of values and then graph the equation. b. Complete this sentence: From the graph, we see that the more trials the rat had, the. c. From the graph, estimate the time it will take the rat to complete the maze on its 32nd trial. d. Interpret the meaning of the \(y\)-intercept.

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(b\) varies directly with \(c\) and inversely with \(d^{2}\). If \(b=5\) when \(c=2\) and \(d=4\), find \(b\) when \(c=36\) and \(d=2\).

Express each direct variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 1. (OBJECTIVE 1) \(r\) varies directly with \(s .\) If \(r=21\) when \(s=6,\) find \(r\) when \(s=12\).

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(y\) varies directly with \(x\) and inversely with \(z .\) If \(y=1\) when \(x=3\) and \(z=7,\) find \(y\) when \(x=8\) and \(z=10\).
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