Problem 57 - Elementary and Intermediate Algebra

Question

Determine whether the given number is a solution of the given equation. $$ 93 ; a-28=75 $$

Step-by-Step Solution

Answer

93 is not a solution.

Step 1 - Understand the Equation

The given equation is written as: \[ a - 28 = 75 \] We need to determine whether the number 93 is a solution to this equation.

Step 2 - Substitute the Given Number

Replace the variable \( a \) with the given number (93) in the equation. This gives us: \[ 93 - 28 = 75 \]

Step 3 - Perform the Subtraction

Calculate the left-hand side of the equation: \[ 93 - 28 = 65 \]

Step 4 - Compare Both Sides

Compare the result from the left-hand side with the right-hand side value. \( 65 eq 75 \)

Step 5 - Conclusion

Since the left-hand side is not equal to the right-hand side, the number 93 is not a solution to the equation \( a - 28 = 75 \).

Key Concepts

Equation Substitution

Equation substitution involves replacing a variable in an equation with a given number or value. In this exercise, we substitute the variable \( a \) with 93. This leads to a new equation, which helps us check if 93 is the solution. Think of substitution as temporarily changing the equation to see if the new statement still holds true.
You can follow these steps for substitution:

• Identify the variable in the original equation.
• Replace the variable with the given number.
• Proceed with arithmetic operations to simplify the equation.

Solving for a Variable

Solving for a variable means finding the value of the unknown that makes the equation true. In our exercise, the unknown is \( a \), and we need to verify if 93 is its correct value. To do this, we:

• Substitute 93 for \( a \) in the equation \( a - 28 = 75 \).
• Simplify the equation to check if both sides are equal.

If the simplified equation is true, then 93 is the solution. If not, it means 93 is not the solution. This process helps us confirm or refute potential solutions.
The main goal is to isolate the variable on one side of the equation. By carefully manipulating the equation, we can see if our substituted value satisfies the original equation.

Basic Algebra

Basic algebra provides the fundamental rules and operations used to manipulate algebraic expressions and equations. It includes addition, subtraction, multiplication, division, and understanding equality. In this exercise, subtracting 28 from 93 is an example of simple arithmetic within the algebraic process.
To successfully solve algebra problems, these skills are crucial:

• Identify and understand the variables and constants.
• Use arithmetic operations to manipulate equations.
• Keep equations balanced by performing the same operation on both sides.
• Evaluate whether the final results satisfy the original equation.

Basic algebra serves as the building blocks for more advanced math concepts and problem-solving strategies. By mastering these essentials, solving more complex problems becomes easier and more intuitive.