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Chapter 2: Problem 59

Solve the equation \(3 x+2=14,\) and graph the solution set on a number line.

Short Answer

Expert verified
The solution is \(x = 4\).

Step by step solution

01

- Isolate the variable term

Start by isolating the term with the variable on one side of the equation. Subtract 2 from both sides of the equation: \[3x + 2 - 2 = 14 - 2\]This simplifies to: \[3x = 12\]
02

- Solve for the variable

Divide both sides of the equation by 3 to solve for \(x\): \[ \frac{3x}{3} = \frac{12}{3} \]This simplifies to: \[ x = 4 \]Therefore, the solution to the equation is \(x = 4\).
03

- Graph the solution on a number line

Draw a horizontal line and place numbers on it. Locate the number 4 on the number line and mark it with a dot. This shows that \(x = 4\) is the solution to the equation.

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Key Concepts

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

isolate the variable
To solve a linear equation, the first crucial step is to isolate the variable, meaning you want to get the term with the variable alone on one side of the equation. For instance, in the equation 3x + 2 = 14, the term with the variable is 3x. We start by removing the constant term (in this case, +2).
You can do this by subtracting 2 from both sides of the equation:
\[3x + 2 - 2 = 14 - 2\]
This simplifies our equation to:
\[3x = 12\]
By isolating the variable term, we've simplified the equation and made it easier to solve for the variable.
graphing solutions
Once you have found the solution to a linear equation, it is helpful to graph the solution to visualize it. In our example, we found that x = 4. To graph this:
  • Draw a horizontal line, which will serve as our number line.
  • Mark a few points on this line, like -2, 0, 2, 4, 6, etc., to give you a range of numbers.
  • Locate the number 4 on your number line and mark it with a dot. This indicates the solution to the equation.

By graphing the solution, you get a visual representation which can make it easier to understand and check your work.
number line
A number line is an essential tool in math. It helps in visualizing numbers and the relationships between them. Here鈥檚 how to draw and use a number line effectively:
Start by drawing a horizontal line and marking a zero in the center. From there, you can mark equal spaces to represent positive and negative integers. For instance, you could mark points for -2, -1, 0, 1, 2, and so on. For our graphing of the solution x = 4, you would locate and mark the point 4 on this line.
Using a number line:
  • Visualizes solutions and helps understand placement and magnitude of numbers.
  • Is useful for addition and subtraction of integers.
  • Helps in understanding inequalities and intervals.

Utilizing a number line makes math clearer and more intuitive.

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