Problem 56 - College Algebra and Calculus: An Applied Approach

Question

A student states that the equation \(-3(x+2)=-3 x+6\) is an identity. Describe and correct the student's error.

Step-by-Step Solution

Answer

The student's error lies in the incorrect distribution of the \(-3\). After correct distribution, the identity should be \(-3(x+2)=-3x-6\), not \(-3(x+2)=-3x+6\).

Step 1: Expand the Left Hand Side (LHS)

Begin with the left-hand side of the equation. To simplify, first distribute the \(-3\) to the terms inside the parentheses, resulting in \(-3x-6\).

Step 2: Simplify the Right Hand Side (RHS)

The right-hand side of the equation is already simplified to \(-3x+6\).

Step 3: Compare LHS and RHS

After simplifying, compare both sides of the equation. Here, it is clear that \(-3x-6\) does not equal \(-3x+6\). Therefore, this equation is not an identity.

Step 4: Correct the Student's Error

To correct the student's error, clarify that when \(-3\) is distributed to the terms inside the parenthesis, it also affects the sign of \(+2\), creating a \(-6\)NOT a \(+6\). Therefore, the correct identity after distributing would be \(-3(x+2)=-3x-6\).

Key Concepts

Identity Equation

An identity equation is a fascinating concept in algebra that holds true for all values of the variable involved. In simpler terms, no matter what number you substitute for the variable, both sides of the equation will always be equal. A classic example would be \(x + 2 = 2 + x\), which simplifies to a tautology, true for any value of \(x\).

However, not all equations are identities. An identity equation essentially becomes a statement of equivalence between two expressions. This means:

• The left-hand side (LHS) and right-hand side (RHS) expressions can appear different at first glance, but after simplification, they resolve to the same expression.
• Identity equations are fundamental in establishing equality across different expressions, highlighting the algebraic flexibility.

In our exercise, if we expand and simplify both sides and find any differences, the equation is not considered an identity, as was the case here.

Distributive Property

The distributive property is a key property of multiplication over addition or subtraction, which simplifies expressions and equations. The basic idea is that a term outside the parentheses multiplies each term inside the parentheses. For example, in the expression \(a(b + c)\), distributive property allows you to transform it into \(ab + ac\).

In this exercise, the distributive property was applied to \(-3(x+2)\), where the \(-3\) multiplies both \(x\) and \(2\). This leads to the simplified form \(-3x - 6\).

• Distribute the multiplier to each term inside the parentheses, adjusting the signs accordingly.
• Ensure that when distributing, each term is accurately multiplied, including adjusting negative signs as necessary.

Understanding this property not only helps in simplifying equations but also in verifying the equality of expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and mathematical operations. These expressions form the core of any algebraic equation and are manipulated using basic arithmetic operations and algebraic properties to solve equations.

In the given exercise, we deal with expressions like \(-3(x+2)\) and \(-3x + 6\). Here, one must skillfully apply algebraic knowledge to simplify or transform expressions effectively.

• Identify each component: variables, coefficients, and constants.
• Apply operations and properties such as distributive, commutative, and associative to transform or simplify expressions.
• Ensure accuracy in sign operations, as this is a common area for errors.

Mastering the manipulation of algebraic expressions allows for greater proficiency in solving more complex algebraic equations and tackling diverse mathematical challenges.