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Chapter 5: Problem 52

Solve and check. $$\frac{1}{2} a-3=1$$

Short Answer

Expert verified
The solution to the equation is \(a=8\).

Step by step solution

01

Addition

Add 3 on each side of the equation, this will cancel out the -3 on the left side. Doing so gives: \( 0.5a = 4 \)
02

Eliminating the Fraction

To isolate \(a\), multiply both sides of the equation by 2, which is the reciprocal of 0.5; \(0.5 \times 2a= 4 \times 2\). This simplifies to \(a=8\)
03

Checking the Solution

Substitute \(a=8\) back into the original equation, which is \(\frac{1}{2} \times 8 -3 =1\). After performing the operations, it simplifies to 1=1, which is true. Hence, the solution \(a=8\) is correct.

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Key Concepts

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

Fractions
Fractions are a way to represent portions of a whole, using a numerator and a denominator. In the exercise, we have a fraction \( \frac{1}{2} a \), implying half of \( a \). Working with fractions can initially seem daunting, but it becomes much easier with practice.

When solving equations with fractions, it's helpful to eliminate fractions early in the process to simplify calculations. For instance, multiplying the entire equation by the reciprocal of the fraction's coefficient, which in this case is 2, can help clear the fraction:
  • The fraction here is \( \frac{1}{2} \), equivalent to 0.5.
  • Multiplying both sides of an equation by 2 removes the fraction, simplifying the problem.
This step turns the fractional equation into a more straightforward one, \( 0.5a = 4 \), allowing us to work with whole numbers.
Equation Solving
Solving an equation is like finding the key to a puzzle. You want to isolate the variable, which in this context is \( a \), to find its value that makes the equation true.

Let's break down the solution method:
  • First, identify and eliminate constants. In the exercise, '3' is subtracted from one side, so we add 3 to both sides to cancel it out. It keeps the equation balanced and results in \( 0.5a = 4 \).
  • After handling constants, take care of coefficients. If the variable is multiplied by a fractional coefficient, multiply both sides by its reciprocal to isolate the variable, simplifying the equation.
Once these steps are followed carefully, you can find the solution. In this case, \( a = 8 \). Solving requires logical operations that keep the equation balanced at every step.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. The goal is to put the equation into a form where the solution becomes obvious.

Let's discuss the core techniques:
  • Addition or subtraction is used to move terms across the equation, such as adding 3 to both sides in the exercise.
  • Multiplication or division clear out coefficients or fractions, as shown by multiplying both sides by 2.
Besides these operations, substitute the solution back into the equation to verify its correctness. Here, substituting \( a = 8 \) back into the original equation confirms that both sides equal, proving that our solution is valid. These manipulations are crucial for efficiently solving algebraic equations.

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Most popular questions from this chapter

A shopping mall has a moving sidewalk that takes shoppers from the shopping area to the parking garage, a distance of 250 ft. If your normal walking rate is \(5 \mathrm{ft} / \mathrm{s}\) and the moving sidewalk is traveling at \(3 \mathrm{ft} / \mathrm{s}\), how many seconds would it take for you to walk from one end of the moving sidewalk to the other end?

Solve and check. $$3.47 a=7.1482$$

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