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Chapter 2: Problem 1

Justify each step. $$\begin{aligned} &4 x-5=-2\\\ &4 x=3\\\ &x=\frac{3}{4} \end{aligned}$$

Short Answer

Expert verified
The solution to the equation \(4x - 5 = -2\) is \(x = 3/4\) or \(0.75\).

Step by step solution

01

Identify the Original Equation

The original equation is \(4x - 5 = -2\). In this equation, the goal is to solve for the variable \(x\).
02

Isolate the x-variable

To isolate \(x\), add \(5\) to both sides of the equation to remove it from the left side. This results in a simplified equation, \(4x = 3\). Adding \(5\) to \(-2\) transforms it into \(3\). Now, \(x\) is only multiplied by \(4\), so it's one step closer to being isolated.
03

Solve for x

Now, to fully isolate \(x\), divide both sides of the equation by \(4\). This leaves \(x\) by itself on one side of the equation. The solution to the equation is then \(x = 3/4\). Dividing \(3\) by \(4\) results in a decimal value of \(0.75\).

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Key Concepts

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

Solving Equations
In Algebra, solving equations involves finding the unknown value of a variable that makes a given equation true. The process typically requires a step-by-step approach to manipulate the equation and isolate the variable. Consider solving the equation \(4x - 5 = -2\). The objective here is to find the value of \(x\) that satisfies this equation. The key to solving equations is to perform the same operation on both sides, keeping the equation balanced.

  • Understand the Objective: Identify what you are solving for. In this case, \(x\) is the variable.
  • Maintain Balance: Whatever you do to one side, do to the other. This keeps the equation equivalent.
  • Perform Operations: Use addition, subtraction, multiplication, or division as needed.

Practice and familiarity with different types of equations can make this process smoother and more intuitive.
Equation Transformation
Equation transformation is a crucial skill that involves changing the form of an equation without altering its solutions. It is central when solving equations, as it allows us to systematically simplify the problem. For instance, to transform our example \(4x - 5 = -2\), we added \(5\) to both sides to obtain \(4x = 3\).

Here are some common transformations:
  • Addition/Subtraction: Used to move terms from one side to the other, like adding \(5\) to eliminate \(-5\).
  • Multiplication/Division: Helpful to simplify coefficients, as seen by dividing by \(4\) to isolate \(x\).
  • Factorization: For more complex equations, breaking down parts into manageable factors.
Understanding how to transform equations allows you to peel back the layers of complexity, simplifying your task down to solving for the variable.
Variable Isolation
Variable isolation is the method used to get the variable alone on one side of the equation. This is the final step in solving equations. In our scenario, the step after transforming it to \(4x = 3\) involves dividing each side by \(4\) to achieve \(x = \frac{3}{4}\). The essence of variable isolation:

  • Identify Terms with the Variable: Start by ensuring the variable is only on one side.
  • Eliminate Coefficients: Divide or multiply to remove coefficients that multiply the variable.
  • Simplify: Combine like terms and simplify the equation as needed to isolate the variable completely.
Effective isolation equips you with the solution, ready to validate or use in further contexts.

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Most popular questions from this chapter

Tell whether each statement is true or false. Then write the converse and tell whether it is true or false. \(n>5\) only if \(n>7\).

Rewrite each pair of conditionals as a biconditional. If \(m \angle A O C=180,\) then \(\angle A O C\) is a straight angle. If \(\angle A O C\) is a straight angle, then \(m \angle A O C=180\).

Provide a counterexample to show that each statement is false. You may use words or a diagram. If a four-sided figure has four right angles, then it has four congruent sides.

Use the given information to write an equation and solve the problem. Find the measure of an angle that is half as large as its complement.

Tell whether each statement is true or false. Then write the converse and tell whether it is true or false. Write a definition of congruent angles as a biconditional.
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