Problem 42 - Prealgebra

Question

Formulate a problem situation for the equation \(x+7=20\)

Step-by-Step Solution

Answer

Sarah has 7 apples and wants 20; she needs to buy more to reach her goal.

Step 1: Identify the Equation Elements

The equation given is \(x + 7 = 20\). Here, \(x\) represents the unknown value we need to solve for, and the numbers 7 and 20 are constants. The equation states that some unknown number plus 7 equals 20.

Step 2: Define a Real-world Scenario

Think of a situation where you start with a number or quantity, add another number to it, and end up with a total. For example, consider a scenario where you already have a number of items and need to buy more to reach a total of 20 items.

Step 3: Construct the Problem Situation

Create a story or scenario based on the real-world context. For instance, "Sarah has 7 apples and wants to have a total of 20 apples. How many more apples does she need to purchase to reach her goal?"

Step 4: Linking Back to the Equation

In the situation, "how many more apples Sarah needs" can be represented by \(x\). The equation \(x + 7 = 20\) models this scenario because combining Sarah's current apples (7) with the apples she needs to buy (\(x\)) will total 20.

Key Concepts

Solving Equations

Equations are like mathematical statements that express equal relationships between two expressions. In the equation, \(x + 7 = 20\), the task is to find the value of \(x\) that makes both sides of the equation equal. This process is called solving an equation. To find \(x\), you can isolate it by performing inverse operations on both sides of the equation. Here, since \(7\) is added to \(x\), you subtract \(7\) from both sides to solve for \(x\): - Begin with the equation: \(x + 7 = 20\). - Subtract \(7\) from both sides: \[(x + 7) - 7 = 20 - 7\] - Simplify the equation: \(x = 13\). This solution demonstrates how practical and methodical solving an equation can be. Once you understand how to manipulate and balance the equation, you can find the unknown variable effortlessly.

Real-World Applications

Understanding equations is essential because they are everywhere in real life. They help solve real-world problems by converting everyday scenarios into mathematical equations. Consider the equation \(x + 7 = 20\) in a real-world context. Imagine you have 7 apples and need a total of 20. The equation helps us determine how many more apples you need to buy (\(x\)). Equations can model:

• Budgeting for shopping (how much you spend to reach a total amount)
• Measuring quantities in recipes (adding ingredients to reach the desired amount)
• Travel planning (how much distance or time is needed to reach a destination)

These examples show how equations are powerful tools to simplify and solve our daily problems, making complex situations more manageable.

Problem-Solving in Math

Problem-solving in math involves finding solutions to different challenges or questions using logical and analytical thinking. It's about understanding the problem, developing a strategy, and working systematically to find an answer. With the equation \(x + 7 = 20\), the problem-solving process begins by clearly defining the scenario. Here’s a step-by-step approach:

• Understand and identify what the problem is asking: How many more apples does Sarah need?
• Translate the problem into a mathematical equation (\(x + 7 = 20\)).
• Develop a strategy – in this case, use inverse operations to isolate \(x\).
• Substitute and solve the equation.
• Check your solution in the context of the problem to ensure it's reasonable.

This structured method not only helps solve the current problem but also equips students with the skills necessary to tackle other mathematical challenges confidently.