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Chapter 1: Problem 37

Solve the given equations. All numbers are approximate. $$-0.24(C-0.50)=0.63$$

Short Answer

Expert verified
The solution to the equation is \(C = -2.125\).

Step by step solution

01

Distribute -0.24

Start by distributing the distribute -0.24 across the parentheses: \(-0.24 \cdot (C - 0.50) = -0.24C + 0.12\). So we rewrite the equation as:\[-0.24C + 0.12 = 0.63\]
02

Subtract 0.12 from Both Sides

Next, isolate the term with the variable by subtracting 0.12 from each side of the equation:\(-0.24C + 0.12 - 0.12 = 0.63 - 0.12\).This simplifies to:\[-0.24C = 0.51\]
03

Divide by -0.24

To solve for \(C\), divide both sides of the equation by -0.24:\(\frac{-0.24C}{-0.24} = \frac{0.51}{-0.24}\).This simplifies to:\[C = -2.125\].

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Key Concepts

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

Equation Solving
Equation solving is a fundamental aspect of algebra. It involves finding the value of the variable that makes the equation true. In this example, we need to determine the value of \(C\) that satisfies the equation \[-0.24(C - 0.50) = 0.63\].To solve equations:
  • Understand the equation completely.
  • Simplify each side of the equation as much as possible.
  • Manipulate the equation step-by-step to isolate the variable.

Equations can be as simple as solving for a single variable or could involve multiple variables or complex algebraic expressions. The goal is always the same: make sure both sides of the equation are equal by the end of your calculations.
Distributive Property
The distributive property is an important algebraic property used to simplify equations and expressions. It lets you distribute a multiplication operation over addition or subtraction inside parentheses. In the original exercise, the equation \[-0.24(C - 0.50)\] needs the distributive property to make it easier to solve.Using the distributive property:\[-0.24 imes (C - 0.50) = (-0.24 imes C) + (-0.24 imes -0.50)\] This operation changes the expression to:\[-0.24C + 0.12\]This technique allows us to remove parentheses and work with simpler terms in the equation, making it easier to solve for the desired variable.
Isolating Variables
Isolating variables is a critical step in solving algebraic equations. The concept involves manipulating the equation so that the variable of interest is alone on one side of the equation.For instance, in the provided equation \[-0.24C + 0.12 = 0.63\], we want to isolate \(C\). We begin by eliminating any constant terms from the side of the variable. This is done by performing the same operation on both sides of the equation. We subtract 0.12 from both sides to simplify to:\[-0.24C = 0.51\]By isolating variables, we systematically reduce the complexity of equations, moving closer to the solution step-by-step.
Division in Algebra
Division in algebra is used to solve for a variable after simplifying an equation. Once we have reached a point where the variable is isolated with a coefficient, we divide to solve for the variable.In the previous steps, we had the equation:\[-0.24C = 0.51\]To solve for \(C\), divide both sides by \(-0.24\):\[C = \frac{0.51}{-0.24}\]This division simplifies to \[C = -2.125\].Division is essential because it helps finalize the solution. Just remember to divide carefully, keeping track of positive and negative signs to ensure the correct solution is achieved.

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Most popular questions from this chapter

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