Problem 46 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ r+30=70 $$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 40 \)

Step 1: Understand the structure of the equation

In the given task, \( r + 30 = 70 \) is a simple linear equation, where 'r' is the variable that needs to be found. This equation demonstrates that some number (r), when added to 30, gives the sum 70.

Step 2: Use inverse operations

To find the value of 'r' we can use inverse operations. The inverse operation of addition is subtraction. So, subtract 30 from both sides of the equation: r = \( 70 - 30 \)

Step 3: Solve the equation

Perform the subtraction on the right-hand side of the equation using mental math. Hence, 'r' equals to \( 40 \).

Key Concepts

Linear Equations

Linear equations are one of the foundational concepts in algebra. They involve expressions that can be set equal to each other. These equations have a characteristic format, usually represented by variables and constants. For example, in the equation \( r + 30 = 70 \), 'r' is the variable and 30 is a constant. Linear equations are called "linear" because if we were to graph them, they would create a straight line. The critical feature of linear equations is their simplicity, which allows us to find a specific solution for the variable.

• The variable (e.g., 'r') represents an unknown number that you are trying to find.
• Constants (e.g., 30 and 70) are numbers with known values.
• We manipulate these components to isolate the variable and solve the equation.

Understanding linear equations paves the way for tackling more complex algebraic problems.

Inverse Operations

Inverse operations are mathematical tools used to solve equations effectively. They help "undo" or reverse operations, making it easier to isolate the variable. For example, the operation of addition is undone by its inverse operation, subtraction.In the example equation \( r + 30 = 70 \), we want to find the value of the variable 'r'. Since 30 is added to r, we use subtraction to remove 30 and solve the equation:- Subtract 30 from both sides: \[ r = 70 - 30 \]Performing this inverse operation helps us get closer to the solution by simplifying the equation.

• Inverse operations are crucial for maintaining the balance of the equation.
• They allow us to isolate the variable on one side of the equation.
• Common pairs include addition/subtraction and multiplication/division.

Using inverse operations correctly leads directly to the solution of the linear equation.

Solving Equations

Solving equations is the process of finding the unknown variable value that makes the equation true. It's essentially like solving a puzzle; you're searching for the missing piece.To solve linear equations, especially with mental math, you can visualize the operations needed to uncover the value of the variable. In our example, once we perform the inverse operation of subtraction, we can easily calculate using mental math:- Calculate: \[ r = 70 - 30 \] This simplifies to \( r = 40 \).

• Solving equations often requires practice with simple arithmetic operations.
• Mental math quickly facilitates solving straightforward linear equations.
• Understanding each step enhances your ability to tackle more complex problems.

By mastering the process of solving equations, you build confidence in algebra and improve your mathematical skills.