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Chapter 3: Problem 58

Solve each equation. $$4+3 c=2(c+2)$$

Short Answer

Expert verified
The solution to the equation is \(c = 0\).

Step by step solution

01

Distribute on the Right Side

Distribute the 2 across the expression \(c + 2\) on the right side of the equation. This means multiplying each term inside the parentheses by 2.\[2(c + 2) = 2c + 4\]
02

Rewrite the Equation

Replace the right side of the equation with the distributed result obtained from Step 1.\[4 + 3c = 2c + 4\]
03

Simplify by Eliminating Like Terms

Subtract 2c from both sides of the equation to eliminate the 2c term on the right side.\[4 + 3c - 2c = 4\]This simplifies to:\[4 + c = 4\]
04

Solve for c

Subtract 4 from both sides to isolate the variable \c\.\[c = 4 - 4\]This simplifies further to:\[c = 0\]

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Key Concepts

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

Equation Solving
Solving equations is a fundamental aspect of algebra. It involves finding the value of variables that make the equation true. In simple terms, an equation is a mathematical statement that asserts the equality of two expressions. To solve an equation means to find the value of the unknown variable that satisfies this equality.
When tackling an equation, it's important to ensure that any operations done to one side of the equation are also done to the other. This maintains the balance of the equation.
  • Identify the equation type: Is it linear, quadratic, or another form?
  • Determine the operations needed to isolate the variable.
  • Perform operations systematically to find the solution.
Once these steps are systematically followed, you'll be well on your way to successfully solving any equation you encounter!
Distribution Property
The distribution property is a key algebraic property used to simplify expressions and equations. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then adding the results. For example, with the equation \(2(c + 2)\):
  • Multiply the term outside the parentheses by each inside term: \(2 \times c\) and \(2 \times 2\).
  • The result is \(2c + 4\).
This property is particularly useful for removing parentheses and simplifying equations, making it easier to isolate the variable and solve the equation.
Isolation of Variables
Isolating variables means rearranging an equation so that the unknown variable stands alone on one side of the equation. This is achieved by performing inverse operations to eliminate terms from the variable's side. Take the simplified equation \(4 + c = 4\) as an example.
To isolate \(c\):
  • Subtract \(4\) from both sides.
  • This step effectively cancels out the \(4\) on the left, leaving \(c\) by itself.
Performing these operations step by step ensures the variable is isolated correctly, leading to the solution. This technique is essential in solving equations effectively and accurately in algebra.

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