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Chapter 1: Problem 30

Solve and check. $$470+c=611$$

Short Answer

Expert verified
Question: Solve the linear equation 470 + c = 611 and check the solution. Answer: c = 141

Step by step solution

01

Isolate the variable

To solve for c, we need to isolate the variable on one side of the equation. To do this, we need to eliminate the number (in this case, 470) adjoining the variable. Subtract 470 from both sides of the equation: $$470+c-470=611-470$$
02

Simplify the equation

Now simplify the equation by performing the subtraction on both sides: $$c=141$$
03

Check the solution

To check the solution, plug the value of c back into the original equation and see if both sides are equal: $$470+141=611$$ $$611=611$$ Since both sides of the equation are equal, our solution, $$c=141$$, is correct.

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Key Concepts

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

Isolate the Variable
In a linear equation like \(470 + c = 611\), the first step is to isolate the variable, in this case, \(c\). Isolating the variable means getting \(c\) on one side of the equation without any other numbers attached to it. This allows you to find the exact value of \(c\).
To isolate \(c\), you must remove any numbers on the same side of the equation as \(c\).
  • Identify the number you need to remove: in this equation, it鈥檚 470.
  • Use the opposite operation to eliminate this number: because 470 is added to \(c\), you'll need to subtract 470 from both sides of the equation.
When you do this, the equation transforms from \(470 + c = 611\) to \(c = 611 - 470\).
By performing the subtraction, you simplify the equation to \(c = 141\). This method of isolating the variable is a crucial skill in solving any kind of algebraic equation.
Checking Solutions
After finding the value of \(c\), it's important to check your solution to ensure it is correct. Checking solutions involves substituting back the found value into the original equation and verifying if both sides remain equal.
  • Take the original equation: \(470 + c = 611\).
  • Substitute the solution \(c = 141\) back into the equation: \(470 + 141\).
  • Perform the addition: if the result (611) matches the right-hand side of the original equation, the solution is correct.
In this case, \(470 + 141 = 611\), and since this is equal to the right side of the equation, it confirms that \(c = 141\) is indeed the correct solution. This checking step is vital as it confirms the accuracy of your solution, ensuring you have not made any errors.
Simplification of Equations
Simplification is a crucial part of solving linear equations. Once you have isolated the variable, simplifying the equation allows you to clearly see the solution. Simplification involves performing the necessary arithmetic operations to reduce the expression to its simplest terms.
In the equation \(470 + c = 611\), after isolating \(c\), you end up needing to perform the subtraction \(611 - 470\).
  • Ensure all arithmetic is correct. Mistakes in basic operations can lead to incorrect results.
  • Whenever you simplify, double-check each step to maintain accuracy.
Simplifying may seem straightforward, but it's important for avoiding mistakes and ensuring that the final answer is as clear and accurate as possible. In this case, simplifying gave us \(c = 141\), which is the exact and simplest form of the solution.

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