Problem 18 - Algebra 1: Concepts and Skills

Question

You put some money in your pocket. You spend \(\$ 4.50\) on lunch. You have \(\$ 7.50\) in your pocket after buying lunch. Solve the equation. What does the solution mean?

Step-by-Step Solution

Answer

You had $12.00 in your pocket before buying the lunch.

Step 1: Formulate the Problem as an Equation

Assume the initial amount of money is represented by x. From the problem, we know that x (initial amount) - $4.50 (amount spent) = $7.50 (amount left). So, the equation becomes \(x - 4.50 = 7.50\)

Step 2: Solve the Equation for x

To solve the equation, we need to isolate x, to find its value. We do this by adding $4.50 to both sides of the equation. This gives \(x = 7.50 + 4.50\)

Step 3: Compute the Value of x

The value of x is thus the sum of $7.50 and $4.50. Hence, \(x = 12.00\)

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In our problem, we had to find out how much money was initially in the pocket. This unknown value was represented by the variable \( x \). We used an equation to express the relationship between the amount in the pocket before and after spending \( \$4.50 \) on lunch. Solving the equation \( x - 4.50 = 7.50 \) required us to isolate \( x \) on one side. We did this by adding \( 4.50 \) to both sides. Such operations help to "undo" the subtraction on the left. Once \( x \) was isolated, we simply calculated the sum of the terms on the right to find that \( x = 12.00 \). This process of rearranging and simplifying until the unknown stands alone is fundamental in solving equations.

Linear Equations

Linear equations are the simplest type of algebraic equations. They form a straight line when plotted on a graph. A typical linear equation takes the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. In our example, \( x - 4.50 = 7.50 \), we are dealing with a linear equation.

• The variable \( x \) was unknown money before the purchase.
• The term \(-4.50\) is a constant representing how much was spent.
• \( 7.50 \) is the remaining money after lunch.

In short, linear equations like this one model simple relationships and can be solved with straightforward steps like addition, subtraction, multiplication, or division, involving only constants and the variable.

Mathematical Operations

Mathematical operations are actions that we perform to solve problems. The basic operations include addition, subtraction, multiplication, and division. In the context of linear equations, these operations help to manipulate expressions and find the value of variables. In our initial step, subtraction was used to describe the spending action: \( x - 4.50 \). When solving equations, it’s often necessary to perform the same operation on both sides of the equation to keep it balanced:

• Addition was then used to solve for \( x \) by adding \( 4.50 \) to both sides of the equation, effectively canceling out the subtraction that originally defined the problem.
• This operation allowed us to find the initial amount in our pocket.

Understanding and applying these basic operations is key to solving any algebraic equation efficiently.