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Chapter 5: Problem 25

Solve and check. $$x+5=7$$

Short Answer

Expert verified
The solution to the equation is \(x = 2\).

Step by step solution

01

Isolate x

Subtract 5 from both sides of the equation in order to isolate x. This gives \(x = 7 - 5\).
02

Simplify the equation

Simplify the right hand side to find the value of x. Thus, \(x = 2\).
03

Check the solution

Substitute x = 2 into the original equation to ensure it's correct. When you substitute 2 for x into the equation \(x + 5 = 7\), the equation simplifies to \(7 = 7\), which is true, so the solution is correct.

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Key Concepts

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

Checking Solutions
After calculating the solution to an equation, it's important to verify its correctness. This process is referred to as checking solutions. To check a solution, substitute the value you found for the variable back into the original equation. For example, if you solved the equation \(x + 5 = 7\) and found \(x = 2\), you would substitute 2 back into the equation in place of \(x\). This gives \(2 + 5 = 7\). If both sides of the equation are equal after substitution, the solution is verified. Always:
  • Substitute the solution into the original equation.
  • Simplify both sides of the equation.
  • Ensure both sides are equal to confirm the solution is correct.
Checking solutions is a crucial step to confirm that you have not made any errors during solving.
Isolating Variables
Isolating variables is a fundamental step in solving equations, as it helps you find the value of the unknown variable. The goal of isolating a variable is to get it alone on one side of the equation. For instance, in the equation \(x + 5 = 7\), the variable \(x\) needs to be isolated for you to determine its value.Here's how you can isolate a variable:
  • Identify the term that is on the same side of the equation as the variable, but does not include the variable (e.g., +5 in the example).
  • Apply the opposite operation to both sides of the equation to remove this term (subtract 5 from both sides).
  • Simplify the equation to ensure the variable is the only term on one side.
In this example, subtracting 5 from both sides of \(x + 5 = 7\) gives \(x = 2\). Isolating variables makes equations much easier to solve.
Substitution Method
The substitution method is not only used for checking solutions but also as a technique in solving systems of equations, where multiple equations are involved. In our simple example, substitution is used to verify an equation's solution.The basic idea is:
  • Take one value from an equation or solution.
  • Replace the variable in another part of the equation with this value.
  • Ensure the remaining expression holds true.
In our example, after finding \(x = 2\), we substitute 2 back into the original equation \(x + 5 = 7\). The new equation, \(2 + 5 = 7\), is true, confirming our solution. Substitution is a valuable method, especially in complex equations, for both finding and verifying solutions.

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