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Chapter 3: Problem 3

Solve the equation. $$-4=3 x-4 x$$

Short Answer

Expert verified
x = 4.

Step by step solution

01

Simplify the Equation

The equation given is \[-4 = 3x - 4x\].Start by combining like terms on the right-hand side.\[3x - 4x = -1x\].So, the equation simplifies to \[-4 = -x\].
02

Solve for x

Now, we need to solve for \(x\). The simplified equation is \[-4 = -x\].To isolate \(x\), divide both sides by \(-1\).\[x = \, \frac{-4}{-1} = 4\].

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Key Concepts

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

Equation Solving
Solving equations can initially feel intimidating, but once you understand the process, it becomes like solving a puzzle. The goal is to find values that make the equation true. Think of it as balancing a scale - whatever you do to one side of the equation, you must do to the other.

Equation solving follows some key steps:
  • Simplify each side of the equation if needed. This makes it easier to focus on one part at a time.
  • Use inverse operations to move terms from one side to the other. If something is added, subtract it; if something is multiplied, divide it.
  • Keep your goal in mind: isolate the variable, which in this case is represented by a letter like \(x\).
In our particular example, we rearranged the terms to solve for \(x\). By isolating \(x\) on one side, we determined its value.
Combining Like Terms
Combining like terms is a fundamental skill used in algebra to simplify equations or expressions. Like terms have the same variable raised to the same power. For example, \(3x\) and \(-4x\) are like terms because they both contain the variable \(x\).

When you combine like terms, add or subtract their coefficients while keeping the variable part unchanged. In our example:
  • We started with \(3x - 4x\).
  • We combined them to get \(-1x\).
This helps simplify the equation, making it easier to solve. Simplification often reveals the clear path to the solution, which is why it's crucial in equation solving. Once the like terms are combined, you're closer to isolating the variable.
Negative Numbers
Working with negative numbers is key to mastering algebra. Negative numbers can initially seem tricky because they involve a direction or value below zero, represented by the \(-\) sign. Here are some tips on handling them:

  • When you multiply or divide two negative numbers, the result is positive. But multiply or divide a negative and a positive, and the result is negative.
  • Subtracting a negative number is the same as adding its positive counterpart.
In the example given, dealing with negative numbers was crucial. The step involving \(-x\) required us to consider the principles of negatives:
  • Dividing \(-4\) by \(-1\) to solve for \(x\) results in a positive \(x = 4\).
Understanding these rules helps in performing the right operations and avoiding common mistakes with equations.

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Most popular questions from this chapter

Solve the equation. $$4 x-3=5 x-1$$

Amber inherits \(26,000 and decides to invest in two different types of accounts, a savings account paying 2.25% interest, and a certificate of deposit paying 4.25% interest. She decides to invest \)6,200 more in the certificate of deposit than in savings. Find the amount invested in the savings account.

Solve the equation. $$1-8(x-8)=17$$

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Solve the equation. $$7 x+2(x+9)=-9$$
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