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Chapter 1: Problem 32

Solve each equation. Check your solution. \(-14+n=-6\)

Short Answer

Expert verified
The solution is \(n = 8\).

Step by step solution

01

Identify the Variable

We start with the equation \(-14 + n = -6\). Our goal is to solve for the variable \(n\). The variable \(n\) is isolated on one side of the equation along with the constant \(-14\).
02

Isolate the Variable

To isolate \(n\), we need to eliminate \(-14\) from the left side of the equation. We do this by adding \(14\) to both sides. This gives us: \(-14 + n + 14 = -6 + 14\).This simplifies to: \(n = 8\).
03

Verify the Solution

We need to check if \(n = 8\) is the correct solution. Substitute \(n\) with \(8\) in the original equation \(-14 + 8 = -6\).Calculate the left side: \(-14 + 8 = -6\). Both sides of the equation are equal, confirming that the solution is correct.

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Key Concepts

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

Isolating Variables
Isolating variables is a crucial step in solving linear equations. It involves getting the variable, such as \(n\) in our example equation, alone on one side of the equation. In \(-14 + n = -6\), the variable \(n\) is accompanied by the number \(-14\). To isolate \(n\), you need to eliminate any constants, or numbers, that are added or subtracted to the variable.

To do this, perform the opposite operation. Since \(-14\) is subtracted, add \(14\) to both sides of the equation. This cancels out the \(-14\) to zero, leaving the equation as \(n = 8\).

Remember:
  • Identify the operation connecting the constant and the variable.
  • Use the inverse operation to cancel the constant.
  • Whatever you do to one side, do to the other to maintain balance.
Checking Solutions
After isolating the variable, it is important to verify your solution to ensure it is correct. This involves checking if substituting the value you obtained back into the original equation satisfies it or not.

For the given equation \(-14 + n = -6\), we found that \(n = 8\). Let's check this solution by substituting \(8\) back into the equation. When we do, it becomes \(-14 + 8 = -6\).

Evaluate:
  • The left side: \(-14 + 8 = -6\).
  • The right side is already \(-6\).
Since both sides of the equation equal \(-6\), the solution is verified successfully. Always double-check your solutions to avoid mistakes.
Substitution Method
The substitution method can be a useful tool for solving equations, especially when checking your solutions. Once you have a proposed solution, you can substitute it back into the original equation to determine if both sides equate.

For example, after finding \(n = 8\) in the equation \(-14 + n = -6\), substitute \(n\) with \(8\) to check if the equation holds. Rewrite the equation: \(-14 + 8 = -6\). This step re-evaluates the expression with your solution value to verify correctness.

Substitution involves:
  • Replacing the variable with its solved value.
  • Recalculate the equation to confirm both sides match.
  • Ensure the equation remains balanced and solve any arithmetic needed.
This method not only confirms your solution but also reinforces your understanding of variable manipulation and equation balance.

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Most popular questions from this chapter

Solve each equation. Check your solutions. $$ |2 x+7|=15 $$

Solve each inequality. Then graph the solution set on a number line. \(14-8 n \leq 0\)

Mrs. Lucas earns a salary of \(\$ 34,000\) per year plus 1.5\(\%\) commission on her sales. If the average price of a car she sells is \(\$ 30,500\) , about how many cars must she sell to make an annual income of at least \(\$ 50,000 ?\) Write an inequality to describe this situation.

Find the value of each expression. $$ (3+7)^{2}-16 \div 2 $$

Define a variable and write an inequality for each problem. Then solve. Three less than twice a number is at most 5 .
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