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Chapter 3: Problem 6

Simplify these equations. Remember to Please Excuse My Dear Aunt Sally. $$ 3 x-2 x+x-4+3-2=0 $$

Short Answer

Expert verified
x = \(\frac{3}{2}\)

Step by step solution

01

Combine Like Terms

Identify and add together all like terms on the left-hand side of the equation. The like terms in this case are the terms involving 'x' and the constant terms. Hence, combine the terms: \(3x - 2x + x\). This simplifies to \((3 - 2 + 1)x = 2x\). Next, combine the constant terms: \(-4 + 3 - 2\). This simplifies to \(-3\).
02

Simplify the Equation

Now that the like terms are combined, rewrite the equation using the simplified terms: \(2x - 3 = 0\).
03

Solve for x

Isolate the variable 'x' by first adding 3 to both sides of the equation: \(2x - 3 + 3 = 0 + 3\), which simplifies to \(2x = 3\). Next, divide both sides by 2: \(x = \frac{3}{2}\).

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Key Concepts

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

simplifying equations
Simplifying equations is a fundamental step in solving algebra problems. It involves reducing the equation to its simplest form while maintaining equality. Simplifying an equation makes it easier to work with and helps in finding the solution more effectively.
Components of an equation may include variables (like x or y) and constants (like numbers 1, 2, 3). To simplify, we need to:
  • Combine like terms
  • Perform basic arithmetic operations
When dealing with an equation like the one given in the exercise, we start by simplifying inside the equation using the rules of algebra.
combining like terms
Combining like terms is essential for simplifying and solving equations. Terms are considered 'like' if they have the same variable raised to the same power. For example, in the term \(3x - 2x + x - 4 + 3 - 2 = 0\), the like terms are those with 'x' and the constant numbers.
Let's break it down:
  • Identify like terms involving the variable: \(3x, -2x, x\)
  • Add them together: \(3x - 2x + x\), which simplifies to \(2x\)
Next, identify and combine the constants:
  • Find the numbers: \(-4, +3, -2\)
  • Add them together: \(-4 + 3 - 2\), which simplifies to \(-3\)
This step-by-step process helps in reducing the complexity and prepares the equation for the next step.
solving for a variable
The ultimate goal in algebra is often to solve for a variable, meaning to find the value of the variable that makes the equation true. After combining like terms and simplifying, we need to isolate the variable.
Given the simplified equation \(2x - 3 = 0\), our focus is on isolating x. Here are the steps:
  • Add 3 to both sides of the equation to remove the constant term from the left side: \(2x - 3 + 3 = 0 + 3\), which simplifies to \(2x = 3\)
  • Next, divide both sides by 2 to solve for x: \(2x / 2 = 3 / 2\), which gives us \(x = \frac{3}{2}\)
Through these steps, we find that \(x = \frac{3}{2}\). This process demonstrates how to systematically isolate the variable and solve the equation.

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Most popular questions from this chapter

Solve for \(x\) $$ 4 x-(2 x-3)=0 $$

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