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Chapter 0: Problem 2

Solve the equation for \(x\) (mentally, if possible). \(x-3=1\)

Short Answer

Expert verified
The solution for the given equation \(x - 3 = 1\) is \(x = 4\).

Step by step solution

01

Identify the equation

The given equation is \(x - 3 = 1\). Our goal is to solve for \(x\).
02

Add 3 to both sides

To isolate \(x\), we need to get rid of the subtraction of 3. We do this by adding 3 to both sides of the equation: \(x - 3 + 3 = 1 + 3\)
03

Simplify and solve for x

Now, simplify the equation and solve for \(x\): \(x = 4\) So, the solution for the given equation is \(x = 4\).

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Key Concepts

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

Solving Equations
Solving equations is a fundamental aspect of algebra that finds the value of a variable that makes an equation true. In simple terms, an equation is like a balance scale, where what you do to one side, you must do to the other to keep it balanced. For the equation given in the exercise, \( x - 3 = 1 \), our goal is to determine what value of \( x \) will make the equation true. The process involves figuring out how to isolate the variable \( x \) on one side of the equation.
  • Identify what operation is being used with the variable.
  • Apply the inverse operation to both sides to keep the equation balanced.
  • Simplify the equation to find the value of the variable.
In the example, since \( x \) is subtracted by 3, we use addition to cancel out the negative 3, and hence solve for \( x \) easily.
Mathematical Thinking
Mathematical thinking is all about recognizing patterns and understanding the structure of problems. When solving the equation \( x - 3 = 1 \), it is crucial to think about what changes could make both sides of the equation equal.
This involves logical reasoning, envisioning different scenarios, and making decisions based on observed patterns.
  • Consider why adding or subtracting a certain number will help solve the problem.
  • Think about the consequences of each mathematical operation involved.
Being able to solve simple equations like \( x - 3 = 1 \) mentally is an outcome of strong mathematical thinking, which is developed over time with practice and engagement with similar problems.
Algebraic Manipulation
Algebraic manipulation involves performing operations to rearrange and simplify equations to solve for unknown variables. In the going exercise, we used addition on both sides to "undo" the subtraction of 3 from \( x \), allowing us to solve the equation efficiently.
  • Identify the operation affecting the variable and perform the opposite operation.
  • Use manipulation step-by-step to maintain the balance of the equation while transforming it.
  • Regularly check each transformation to ensure that no errors are made.
The ability to manipulate algebraic expressions and equations correctly is an essential skill in algebra because it enables students to handle more complex problems in future learning scenarios.

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Most popular questions from this chapter

Calculate each expression in Exencises giving the answer as \(a\) whole mumber or a fraction in lowest terms.\(3\left(\frac{-2 \cdot 3^{2}}{-(4-1)^{2}}\right)\)

Find all possible real solutions of each equation \(y^{3}+64=0\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(3+\frac{3}{2-9}\)

Solve the equation for \(x\) (mentally, if possible). \(a x+b=c \quad(a \neq 0)\)

Find all possible real solutions of each equation \(2 x^{6}-x^{4}-2 x^{2}+1=0\)
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