Problem 30 - Algebra A Combined Function

Question

Solve each equation. See Examples 6 and \(7 .\) $$ 14 x+7=7(2 x+1) $$

Step-by-Step Solution

Answer

The equation is true for all real numbers; every \( x \) is a solution.

Step 1: Distribute Terms on the Right Side

The original equation is \( 14x + 7 = 7(2x + 1) \). Begin solving by distributing the 7 to both terms inside the parenthesis on the right side. This simplifies to: \( 14x + 7 = 14x + 7 \).

Step 2: Simplify Both Sides

Compare both sides of the equation: \( 14x + 7 = 14x + 7 \). Since both sides are identical, no further simplification is needed.

Step 3: Determine the Solution

Since the left side \( 14x + 7 \) is equal to the right side \( 14x + 7 \) for all values of \( x \), every real number is a solution to this equation.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra and mathematics in general. It involves finding the value(s) of the variable(s) that make the equation true. In simpler terms, you are looking for what number or numbers you can substitute for the variable to balance the equation.
To solve equations efficiently, follow these steps:

• Identify the type of equation you are dealing with. Linear equations, like in our example, have variables raised to the power of one.
• Apply algebraic techniques such as addition, subtraction, multiplication, division, and distribution to both sides of the equation equally to maintain balance.
• Simplify each side of the equation as much as possible.
• Check your solution by substituting it back into the original equation to ensure it holds true.

In some cases, like the one we have here, you may find both sides are identical. This means the equation holds true for any value of the variable, concluding that the solution is every real number. Understanding these steps is crucial in mastering equation solving.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition or subtraction) grouped together. They form the building blocks of equations and differ from equations as they do not have an equality sign.
Here's a quick breakdown of components of an algebraic expression:

• Terms: Individual parts of the expression separated by plus or minus signs. In the expression \( 14x + 7 \), "14x" and "7" are terms.
• Coefficients: The numerical part of a term that is multiplied by the variable. For "14x," the coefficient is 14.
• Variables: Symbols used to represent unknowns, like \( x \).

Understanding algebraic expressions is essential to manipulate and solve equations. By analyzing each term, coefficient, and variable, you can apply the correct mathematical techniques to simplify expressions and solve equations.

Distributive Property

The distributive property is a key algebraic concept used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each individual term by that number, then adding the results.
Take the expression \( 7(2x + 1) \) from our example. Using distributive property, this can be expanded to:

• Multiply 7 by \( 2x \), giving \( 14x \).
• Multiply 7 by 1, giving 7.
• Combine these results to get \( 14x + 7 \).

This technique is especially useful when simplifying expressions with parentheses or when dealing with equations involving variables. By breaking down complex expressions into simpler terms, the distributive property helps you intuitively understand and solve equations.