Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the equation is an identity or a conditional equation. $$ 3(x+2)=2 x+4 $$

Step-by-Step Solution

Answer

The given equation is a conditional equation, not an identity, and it is only valid when \(x=-2\).

Step 1: Expand the equation

Distribute multiplication in the equation \(3(x+2)=2x+4\) to get a simplified form. Therefore, it becomes \(3x+6=2x+4\).

Step 2: Rearrange the equation

Rearrange the equation to get the value of x, on its own, on one side of the equation. In this case, subtract \(2x\) from both sides of the equation. This gives \(x+6=4\).

Step 3: Solve for x

To completely isolate x, subtract 6 from both sides of the equation. This results in \(x=-2\).

Step 4: Identity or Conditional equation

Since our equation is only true for \(x=-2\) and not for all values, this indicates that it's a conditional equation, not an identity. The equation is only true when \(x=-2\), not for all possible values of \(x\).

Key Concepts

Solving Linear Equations

Understanding how to solve linear equations is instrumental for students diving into algebra. A linear equation is a mathematical statement that shows the equality of two expressions, typically with a variable such as x that we need to solve for. Here's a step-by-step guide on tackling these equations:

First, simplify both sides of the equation by eliminating parentheses and combining like terms, using the distributive property if necessary. Next, gather all terms with the variable on one side and the constant terms on the other side by adding or subtracting appropriately. Finally, isolate the variable by performing operations to get it by itself, allowing its value to be determined. In the exercise, after distributing and rearranging, we determined that the solution is x = -2, which denotes a specific solution, thus making it a conditional equation.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to remove parentheses by distributing a multiplied value across terms within the parentheses. For example, considering a(b + c), the distributive property lets us expand this to ab + ac. In our exercise, we applied this property to distribute the 3 across both x and 2 in the expression 3(x+2), resulting in 3x + 6. The beauty of the distributive property lies in its ability to simplify equations into a more manageable form, easing the process of solving for the variable. It's a basic yet powerful tool that algebra students should master early on.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables within them. They are the building blocks used to simplify expressions and solve equations. Common identities include a^2 - b^2 = (a + b)(a - b) and (a + b)^2 = a^2 + 2ab + b^2. In the context of the given exercise, we had to determine if the equation 3(x + 2) = 2x + 4 is an identity or a conditional equation. Since it holds true only for x = -2, it is not an identity. Understanding the difference between identities, which are true for any variable input, and conditional equations, which are true for specific values, is important for grasping fundamental algebra concepts.