Problem 53 - Algebra 1

Question

The bill (parts and labor) for the repair of a car was S458. The cost of parts was \(\$ 339\). The cost of labor was \(\$ 34\) per hour. Write and solve an equation to find the number of hours of labor.

Step-by-Step Solution

Answer

The number of labor hours is approximately \(H\) = 3.5 hours.

Step 1: Define the variables

Let's assume \(H\) to be the number of hours of labor which we need to find out.

Step 2: Formulate the equation

Knowing that the total cost is given by the formula \(Total\) = \(cost\_of\_parts\) + \(cost\_of\_labor\_per\_hour * hours\_of\_labor\), we can substitute the given values to get this equation: 458 = 339 + 34\(H\).

Step 3: Solve the equation

The aim is to isolate \(H\) for us to find out its value. From the equation in step 2, we get 458 - 339 = 34\(H\), which simplifies to 119 = 34\(H\). Dividing both sides by 34 to solve for \(H\), \(H\) = 119/34.

Step 4: Evaluate the result

Check that our solution makes sense in the context of the problem. The number of hours of labor can't be negative, and in this case, is a fraction which is possible when dealing with time. So the solution is reasonable.

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of solving equations in algebra. These are mathematical phrases that can contain ordinary numbers, variables (like x or H), and operators (such as plus, minus, multiply, and divide). The beauty of an algebraic expression is that it can represent a multitude of real-world scenarios. For instance, let's say you're working out the cost of a car repair, and you know the cost of parts and the per hour charge for labor. You could write an expression like this: total cost = cost of parts + (cost of labor per hour * hours of labor).

The key to mastering algebraic expressions is understanding how to manipulate them. This means being able to add, subtract, multiply, and divide on both sides of the expression to solve for the unknown variable. What's important is to perform the same operation on both sides to maintain the balance; this is the core principle behind solving any algebraic equation, which we will delve into next.

Equation Solving

The process of solving an equation is like a treasure hunt - the 'x marks the spot' being your unknown variable. When faced with an equation, your goal is to isolate this variable and 'solve' for it, which means finding the value that makes the equation true. It's a systematic process, where each step is designed to simplify the equation. Following the car repair example, we know that the equation is 458 = 339 + 34H, which is already set up for us. Here's how it's done step by step:

• Identify the terms containing the variable and move all other terms to the opposite side.
• Perform the operations required to isolate the variable, in this case, subtracting 339 from both sides.
• Once the variable term is alone, divide by the coefficient of the variable (the number in front of the variable), here it's 34.
• Verify that the solution makes sense within the context; for instance, hours of labor cannot be negative.

Practice makes perfect in equation solving and understanding the 'why' behind each step is crucial for mastering more complex problems down the line.

Word Problems in Algebra

Word problems often strike fear into the hearts of algebra students, yet they are a vital skill. To tackle a word problem, start by reading the problem carefully and identifying what's being asked for. The next step is to define the variable; in the car repair scenario, we're looking for the hours of labor, so we'll call it H.

Once the variable is set, we translate the written words into an algebraic equation. In our example, the total bill is the sum of the parts cost and the hourly labor cost multiplied by the number of hours worked, leading us to an equation we can solve.

• Read the problem and identify what you need to find.
• Define the variable that represents the unknown in the context of the problem.
• Translate the words into algebraic expressions and create an equation.
• Solve for the variable using standard algebraic methods.
• Interpret the result and ensure it makes sense.

With practice, word problems can become a fun puzzle to solve, integrating real-life context into mathematical equations. Remember, attention to detail is key when it comes to word problems — always double-check the result against the given scenario.