Problem 37 - Elementary Algebra

Question

\(6 x+4 x=x(6+4)\)

Step-by-Step Solution

Answer

The equation \(6x + 4x = x(6+4)\) is true because both sides simplify to \(10x\).

Step 1 - Identify the expression on both sides of the equation

Observe that the left side of the equation is given as: \[6x + 4x\] and the right side is: \[x(6 + 4)\]

Step 2 - Combine like terms on the left side

Add the coefficients of \(x\) on the left side: \[6x + 4x = 10x\]

Step 3 - Simplify the right side

Calculate the expression inside the parentheses on the right side: \[6 + 4 = 10\]. Therefore, the right side becomes: \[x(10) = 10x\]

Step 4 - Compare both sides

Now, compare the simplified expressions from both sides of the equation: \[10x = 10x\]. Both sides are equal, which verifies the equation.

Key Concepts

headline of the respective core concept

Simplifying algebraic expressions involves reducing an algebraic expression to its most concise and manageable form. This usually means combining like terms and performing basic arithmetic operations.
In our original exercise, we started with the equation: \(6x + 4x = x(6 + 4)\).

The first step in simplifying algebraic expressions is to identify and combine like terms. 'Like terms' are terms that have the same variable raised to the same power. In this case, both terms on the left side of the equation, \(6x\) and \(4x\), are like terms because they both contain the variable \(x\) with the same exponent. By adding their coefficients, we simplify \(6x + 4x\) to \(10x\).
This process of combining like terms helps to make the expression simpler and easier to work with. So, after combining like terms, the left side of our equation becomes \(10x\).
Combining like terms effectively reduces the complexity of large algebraic expressions and makes them easier to solve.

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The distributive property is a fundamental concept in algebra which states that multiplying a number by a sum of numbers is the same as doing each multiplication separately.
In mathematical terms, this property is expressed as: \(a(b + c) = ab + ac\).

In our exercise, \(x(6 + 4)\), we can apply the distributive property. To distribute the \(x\) across the sum inside the parentheses, we multiply \(x\) by each term inside the parentheses separately. This gives us: \(x*6 + x*4 = 6x + 4x\).
This step shows that whether we simplify the left side first, or distribute on the right side first, we end up with the same simplified term: \(10x\).
The distributive property helps to break down complex expressions and make algebraic manipulations more straightforward, by ensuring that multiplication across a sum or difference is handled correctly.

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Understanding elementary algebra involves mastering basic operations and principles that form the foundation of algebra.
This includes operations like addition, subtraction, multiplication, and division of algebraic expressions. It's also important to understand properties like the distributive property, commutative property, and associative property.

In our given problem, we used some key concepts from elementary algebra:

• Combining like terms: \(6x + 4x = 10x\)
• Using the distributive property: \(x(6+4) = x*6 + x*4 = 6x + 4x\)

Both of these steps simplify the expressions effectively, making it easier to solve the equation.
By comparing both sides: \(10x = 10x\), you verify the correctness of the algebraic manipulations.
Elementary algebra provides the building blocks for more advanced mathematical concepts and is essential for students to understand thoroughly as they progress in their studies.