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Chapter 2: Problem 12

Solve each equation, and check your solution. $$ x+7=11 $$

Short Answer

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

Step by step solution

01

Isolate the variable

To isolate the variable \(x\), subtract 7 from both sides of the equation \(x + 7 = 11\).\[x + 7 - 7 = 11 - 7\]
02

Simplify both sides

Simplify the equation after subtracting 7 from both sides.\[x = 4\]
03

Check the solution

To ensure the solution is correct, substitute \(x = 4\) back into the original equation and simplify.\[4 + 7 = 11\] Since both sides are equal, \(x = 4\) is indeed the solution.

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Key Concepts

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

Isolating Variables
To solve a linear equation, the first step often involves isolating the variable. This means getting the variable by itself on one side of the equation. For example, in the equation \(x + 7 = 11\), we need to isolate \(x\).

Here鈥檚 how we do it:
  • Identify the term that does not involve the variable. In this case, it is 7.
  • Perform the opposite operation to both sides of the equation to eliminate the term. Since 7 is added to \(x\), we subtract 7 from both sides: \(x + 7 - 7 = 11 - 7\).
  • After subtraction, we get \(x = 4\).
Now, the variable \(x\) is isolated and we have our potential solution, \(x = 4\).
Checking Solutions
Checking your solution is a crucial step. It ensures that the value you have found actually satisfies the original equation.

To check the solution \(x = 4\) for the equation \(x + 7 = 11\):
  • Substitute \(4\) back into the original equation in place of \(x\): \(4 + 7 = 11\).
  • Simplify the left side of the equation: \(11 = 11\).
  • If both sides are equal, the solution is correct. In this case, \(11 = 11\) confirms that \(x = 4\) is indeed the correct solution.
Through this, students can be confident that their steps and calculations are accurate.
Simplifying Equations
Simplifying equations is about making the equation easier to solve by performing operations that reduce its complexity.

For the equation \(x + 7 = 11\):
  • We began by subtracting 7 from both sides to simplify the equation: \(x + 7 - 7 = 11 - 7\).
  • This simplification step results in \(x = 4\), which is much easier to understand and verify.
Simplification often involves basic arithmetic operations like addition, subtraction, multiplication, and division. Always perform the same operation on both sides of the equation to keep the equation balanced.
By regularly simplifying equations, students will find solving them more straightforward and less intimidating.

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Most popular questions from this chapter

Write an equation using the information given in the problem. Use \(x\) as the variable. Then solve the equation. If twice a number is divided by \(5,\) the result is \(4 .\) Find the number.

Convert each decimal to a percent. $$ 0.983 $$

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Solve each problem. 8 is what percent of \(64 ?\)
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