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Chapter 2: Problem 190

Solve the following equations with variables on both sides. $$c=-3 c-20$$

Short Answer

Expert verified
The solution is \( c = -5 \).

Step by step solution

01

- Move Variable Terms to One Side

Add 3c to both sides of the equation to move all the variable terms to one side: \[ c + 3c = -3c - 20 + 3c \]
02

- Simplify Both Sides

Combine like terms on both sides: \[ 4c = -20 \]
03

- Solve for the Variable

Divide both sides by 4 to isolate the variable: \[ c = \frac{-20}{4} \] \[ c = -5 \]

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Key Concepts

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

variables on both sides
When you encounter an equation with variables on both sides, your first goal is to move all the variables to one side of the equation. This helps simplify the problem. Let's consider the equation given:
$$c = -3c - 20$$
To remove the variable from the right side, add its opposite to both sides. Adding 3c to both sides, we get:
$$c + 3c = -3c - 20 + 3c$$
This leaves us with variables on only one side of the equation. Now you can move on to the next step.
combining like terms
After moving the variables to one side, it's time to combine like terms. In our example, after adding 3c to both sides, we get:
$$c + 3c = -3c - 20 + 3c$$
Combining the terms on the left side gives us:
$$4c = -20$$
Like terms are terms that contain the same variables raised to the same power. In this step, we combined 1c with 3c to get 4c. Simplifying the equation in this way makes it clearer and easier to solve.
isolating the variable
The primary goal in algebra is often to solve for a variable by isolating it on one side of the equation. From our earlier steps, we now have:
$$4c = -20$$
To isolate c, you need to undo the multiplication by 4. This is done by dividing both sides of the equation by 4:
$$c = \frac{-20}{4}$$
This simplifies to:
$$c = -5$$
The variable is now isolated and we have our solution. Simple steps like these can help make sure you correctly solve for the variable every time.
elementary algebra
Elementary algebra involves basic mathematical skills and operations often applied to solving equations. In this problem, we used a few fundamental concepts of elementary algebra:
  • Moving variables to one side
  • Combining like terms
  • Isolating the variable

Each of these steps helps in simplifying the equation step-by-step, making complex problems easier to solve. Mastering these basics of elementary algebra is crucial for tackling more advanced mathematical problems in the future.

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