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Chapter 3: Problem 6

Solve the equation. $$7+c=-10$$

Short Answer

Expert verified
The solution to the equation is \(c = -17\).

Step by step solution

01

Subtract 7 from both sides of the equation

To isolate variable 'c', subtract 7 from both sides of the equation. This gives us: \(c = -10 - 7\)
02

Solve for 'c'

Next, compute the right-hand side to solve for 'c': \(c = -17\)

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Key Concepts

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

Isolation of Variables
In algebra, isolating the variable is a fundamental step toward solving linear equations. It involves manipulating the equation to get the unknown variable on one side. This allows for easier computation and final solution.
To isolate a variable, follow these basic ideas:
  • Look at both sides of the equation and identify which terms need to be eliminated or moved.
  • Use inverse operations. For example, if a number is added to the variable, subtract it from both sides.
  • Keep the equation balanced by performing the same operation on both sides. This is crucial as changing one side without doing the same to the other will make the equation invalid.
For the equation given, \(7 + c = -10\), we needed to subtract 7 from both sides to isolate \(c\). This step simplifies the rest of the solution, allowing you to easily find the value of \(c\).
Solving Equations
Once the variable is isolated, you need to solve the equation. Solving equations involves computing the result after isolation, which gives us the value of the unknown.
Solving linear equations usually includes:
  • Performing basic arithmetic operations like addition, subtraction, multiplication, or division.
  • Substituting the known values if necessary to simplify the equation.
  • Checking your solution by plugging it back into the original equation to ensure both sides are equal.
With our solved equation \(c = -10 - 7\), the next step is to perform the subtraction. When you compute \(-10 - 7\), you get \(-17\). Therefore, the solution for \(c\) in the equation is \(-17\). Always make sure to verify your solution.
Integer Operations
Understanding integer operations is critical when solving equations because it helps in calculating accurate results. Integers include whole numbers, positive or negative, and zero. In algebra, operations with integers follow specific rules:
  • Adding a negative number is the same as subtracting the absolute value of that number.
  • Subtracting a negative number equates to adding its absolute value.
  • The result of a positive number minus a negative number becomes more positive.
In our exercise, we handled the operation \(-10 - 7\), which resulted in \(-17\). Here's the rule applied: Subtracting 7 from \(-10\) is like adding \(-7\) because of the negative sign, leading us to the final result of \(-17\). Correct handling of integer operations is necessary for obtaining the right answers in equations.

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