Problem 29 - Prealgebra

Question

During the summer, Ana grew from a height of 65 inches to a height of 68 inches. Define a variable. Then write an equation that can be used to find the increase in height. How many inches did Ana grow?

Step-by-Step Solution

Answer

Ana grew 3 inches over the summer.

Step 1: Define the Variable

Let's define a variable to represent the increase in Ana's height. Let \( x \) be the increase in height in inches. This will help us set up our equation.

Step 2: Set Up the Equation

To find the increase \( x \), we can use the initial and final heights. Ana's initial height was 65 inches, and her final height was 68 inches. Therefore, the equation relating these is \( 65 + x = 68 \).

Step 3: Solve the Equation

Now solve the equation \( 65 + x = 68 \) for \( x \). Subtract 65 from both sides of the equation to isolate \( x \):\[x = 68 - 65\]This gives us \( x = 3 \).

Step 4: Interpret the Result

The solution \( x = 3 \) tells us that Ana grew a total of 3 inches over the summer.

Key Concepts

Understanding Variables

Variables are like placeholders or symbols used in mathematics to represent unknown values. When solving problems, variables allow us to write equations and expressions that describe relationships between numbers. In this exercise, we needed to define a variable to represent Ana's unknown increase in height since the exact number of inches she grew wasn't initially given. We chose "\( x \)" as our variable to symbolize Ana's growth in inches.

Defining variables is essential for solving equations because it turns the problem into an algebraic form that can be manipulated. It makes it easier to understand and solve real-world problems. In dealing with prealgebra equations, always start by identifying what is unknown in the problem, then assign a variable to this unknown. The variable is a starting point for setting up an equation to find the solution. Using variables is a fundamental skill in mathematics, as it forms the basis for more complex algebra and calculus.

Exploring Linear Equations

Linear equations are one of the simplest types of equations you will encounter in prealgebra. They are called "linear" because when you plot them on a graph, they form a straight line. The general form of a linear equation is \( ax + b = c \), where \( x \) is the variable, and \( a \), \( b \), and \( c \) are constants.

In the case of Ana's height growth, we dealt with a linear equation: \( 65 + x = 68 \). Here, "65" represents Ana's initial height, "68" is her final height, and "\( x \)" is the variable indicating the increase in her height.

• Linear equations are straightforward because they require basic arithmetic operations like addition, subtraction, multiplication, and division to solve.
• They can be solved by isolating the variable, typically through inverse operations, like subtracting numbers on both sides as we did in the third step of the solution.

This process highlights the simplicity yet power of linear equations in modeling everyday situations and solving practical problems.

Effective Problem Solving

Problem solving in mathematics involves identifying the problem, understanding the known and unknown quantities, and using appropriate methods to find solutions. Each step in solving Ana's height problem is a perfect example of effective problem solving.

By defining a variable, like \( x \) to represent the increase in Ana's height, we started by isolating the unknown component of the problem. This approach is efficient and prevents confusion, making the problem-solving process smoother.

When working through problems, here are some tips:

• Carefully read the problem to understand what is being asked.
• Determine what information is provided and what needs to be found.
• Translate the real-world problem into a mathematical model using equations and variables.
• Solve the mathematical equation for the unknown variable.
• Always check the solution in the context of the original problem to ensure it makes sense.

By routinely following these steps and practicing consistently, you'll improve your problem-solving skills and handle more complex mathematical challenges with confidence.