Problem 74 - Algebra and Trigonometry

Question

Combine the types of equations we have discussed in this section. Solve equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$2(x+2)+2 x=4(x+1)$$

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Simplify both sides of the equation

Simplify both sides of the equation \(2(x+2)+2x=4(x+1)\) to get \(2x + 4 + 2x = 4x + 4\).

Step 2: Simplify again

Combine like terms on both sides of the equation to get \(4x + 4 = 4x + 4\).

Step 3: Solve the equation

Try to solve the equation for \(x\) by subtracting \(4x\) from both sides, which results in \(0 = 0\).

Step 4: Identify the type of equation

Since we now have \(0 = 0\), which is always true, we find every value of \(x\) is a solution. This results in the equation being an identity.

Key Concepts

Types of Equations

An equation in algebra represents the equality of two expressions. Understanding the types of equations is fundamental as it helps in identifying the solution strategy. There are primarily three types of equations students should be familiar with:

• Identities which always hold true, no matter the value of the variable.
• Conditional equations which are true only for certain values of the variable.
• Inconsistent equations which have no solution as they represent a false statement.

In the given exercise, we see an example of an identity, as it simplifies to a true statement independent of the variable's value.

Identities in Algebra

In algebra, an identity is an equation that is always true, regardless of the values plugged into the variables. For example, the equation \(a + b = b + a\) is an identity because no matter which numbers \(a\) and \(b\) represent, the equation will always be true due to the commutative property of addition.

Identities are useful in simplifying expressions and solving complex problems. The original exercise presents an identity, as every value of \(x\) will make the equation \(0 = 0\) valid.

Conditional Equation

A conditional equation is true only for specific values of its variables, unlike an identity which is true for all possible values. For instance, the equation \(x + 2 = 5\) is conditional because it is only true when \(x = 3\). Solving conditional equations typically means finding the value(s) of the variable(s) that make the equation true.

Students must perform algebraic operations such as factoring, distributing, or combining like terms to isolate the variable and find the solution to a conditional equation.

Inconsistent Equation

An inconsistent equation is one that has no solution because it results in a mathematical impossibility, such as \(5 = 7\). No matter what value you substitute for the variable, the equation will never hold true. Detecting inconsistent equations early can save time, as no further algebraic manipulation will yield a solution.

Understanding that some equations may have no solution helps set realistic expectations when encountering a problem, knowing the possibility exists that the equation may not be solvable.

Combining Like Terms

The process of combining like terms means to simplify an expression or an equation by adding or subtracting terms that have the same variable raised to the same power. For instance, in the expression \(2x + 3x\), we combine the like terms to get \(5x\). It is an essential step in solving equations because it allows us to reduce the equation to a simpler form before proceeding to isolate the variable.

In the context of the exercise, we saw that combining like terms on both sides gave us an identical expression, leading us to conclude that the equation is an identity. This step is sometimes where the type of equation we are dealing with becomes apparent.