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Chapter 3: Problem 2

Solve each equation. $$3 x+4=19$$

Short Answer

Expert verified
The solution is \( x = 5 \).

Step by step solution

01

Isolate the Variable

Start by isolating the variable, which means getting the term with the variable by itself on one side of the equation. To do this, subtract 4 from both sides of the equation:\[3x + 4 - 4 = 19 - 4\]This simplifies to:\[3x = 15\]
02

Solve for the Variable

Now, solve for the variable by dividing both sides by 3 to get the value of \( x \):\[\frac{3x}{3} = \frac{15}{3}\]This simplifies to:\[x = 5\]

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Key Concepts

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

Isolating the Variable
In order to solve linear equations, one of the main steps is to *isolate the variable*. This means getting the variable by itself on one side of the equation. Here is a simple quick explanation on how you would do this:
  • Start by examining the equation to identify the term that contains the variable you need to solve for.
  • Move all other terms to the opposite side of the equation by applying inverse operations (like subtraction if a number is being added to the variable).
  • Continue until only the variable and its coefficient are on one side.

For example, in the equation \(3x + 4 = 19\), the term \(4\) is added to the variable term \(3x\). Subtract \(4\) from both sides to eliminate this term, which results in \(3x = 15\). This is crucial because simplifying the equation in this way prepares it for the next step where you solve for the variable.
Step by Step Solution
Working through a *step by step solution* can make solving equations much simpler and manageable. Here is a breakdown of the approach used in the example exercise:
  • **Step 1**: Isolate the variable as mentioned earlier. In the exercise, this involves subtracting \(4\) from both sides, yielding \(3x = 15\).
  • **Step 2**: Solve for the variable. This is achieved by adjusting the equation so that the variable stands alone. In this particular example, you do this by dividing both sides of the equation by \(3\), giving \(x = 5\).

This structured way of approaching problems makes it easier to track what you are doing at each stage, reducing the potential for errors. Following each step attentively ensures that each stage contributes correctly to finding the solution.
Basic Algebra Concepts
Understanding *basic algebra concepts* is foundational for solving equations efficiently. Here are some key points:
  • *Operations*: Basic operations like addition, subtraction, multiplication, and division are crucial in rearranging and solving equations.
  • *Inverse Operations*: These are operations that reverse or undo each other, such as addition and subtraction or multiplication and division. They help move terms from one side of an equation to the other.
  • *Equation Balance*: An equation must remain balanced, meaning whatever you do to one side, you must do to the other. This balance is key to maintaining equality while isolating or solving for variables.

With these concepts, you'll have a better grasp of how to manipulate equations, making solving them not just something you do, but something you understand. Remember: Practice is vital. The more you solve, the more intuitive these concepts become.

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Most popular questions from this chapter

Solve each equation. $$5(x-4)=4(x+6)$$

Solve each equation. $$\frac{x-1}{5}-\frac{x+4}{6}=-\frac{13}{15}$$

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