Problem 9 - Algebra 1

Question

You have a summer job running errands for a local business. You earn 5 dollar per day, plus 2 dollar for each errand. Write and solve an equation to find how many errands you need to run to earn 17 dollar in one day.

Step-by-Step Solution

Answer

You need to run 6 errands to earn $17 in one day.

Step 1: Formulate the Equation

Let x represent the number of errands. The total money earned in a day, given by a fixed $5 salary plus $2 for each errand, would be expressed in the equation \(5 + 2x = 17\).

Step 2: Simplify the Equation

Subtract 5 from both sides of the equation to isolate the term with x, resulting in the simplified equation \(2x = 17 - 5\), which simplifies further to \(2x = 12\).

Step 3: Solve for x

Divide both sides of the equation by 2 to solve for x, obtaining \(x = 12 / 2\), which simplifies to \(x = 6\).

Key Concepts

Equation Solving

Equation solving is all about finding the value of an unknown variable that makes the equation true. In our exercise, we needed to determine how many errands would allow you to earn $17 in just one day. We set up an equation:

• 5, for the fixed salary.
• 2x, for the additional money earned by running errands.
• 17, the amount you want to achieve.

The equation becomes:
\[5 + 2x = 17\]
Solving this involves moving the terms around to isolate the variable on one side. This process may involve using subtraction, addition, multiplication, or division, depending on the given problem.

Variable Isolation

Variable isolation is the process of rearranging an equation to get the unknown variable by itself on one side. In our example, isolate the variable 'x', which represents the number of errands.
To do this:

• Subtract 5 from both sides:
  \[2x = 17 - 5\]
• Calculate the result:
  \[2x = 12\]

Now, we have brought the equation into a form where the next steps will clearly show the value of the variable.

Basic Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are used to rearrange and simplify equations. Let's use these to solve the equation \(2x = 12\).
To isolate 'x', divide both sides by 2:

• \(x = \frac{12}{2}\)
• This simplifies to \(x = 6\).

These basic operations are fundamental because they allow us to manipulate equations and find the values of unknown variables easily. Understanding these operations will help you solve more complex problems in the future.