Problem 18 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$4(z-2)=2(2 z-4)$$

Step-by-Step Solution

Answer

The equation is an identity and therefore, it is true for all values of \(z\).

Step 1: Distribute

Distribute the terms on both sides of the equation. After distribution, the equation will be \(4z - 8 = 4z - 8\).

Step 2: Simplify

Simplify further by relocating terms. When the \(4z\) is subtracted from both sides of the equation, we have \(-8 = -8\).

Step 3: Check The Result

After simplifying, you will notice that all variables have been eliminated and we are left with a true statement, \(-8 = -8\), which means our original equation is an identity equation and true for all values of \(z\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that requires us to find the value of variables that satisfy given equations. In this particular problem, we start with the equation \(4(z-2)=2(2z-4)\). Our goal is to find values for \(z\) that make both sides of the equation equal. To solve it, we often follow these steps:

• Distribute and Simplify: Begin by expanding both sides of the equation using the distributive property, which allows us to rewrite equations in simpler forms.
• Relocate Terms: Adjust the equation by adding or subtracting variables and constants, so the variable terms are on one side of the equation.
• Check: Verify your solutions by substituting back into the original equation to ensure they hold true.

Here, when we simplify our equation, we noticed that the variables cancel out and we are left with a true statement, helping us conclude our findings.

Distributive Property

The distributive property is a vital tool in algebra that lets you expand expressions and simplify equations. It states that you can distribute a multiplier across terms within parentheses, enabling us to handle complex algebraic expressions more easily. For our exercise, we applied the distributive property as follows:

• On the left: \(4(z-2)\) becomes \(4z - 8\).
• On the right: \(2(2z-4)\) becomes \(4z - 8\).

By distributing, we clearly see how each term outside the parentheses affects every term inside, allowing us to line up terms for further operations. This step simplifies our original equation and is key in making it more manageable for subsequent simplifications.

Identity Equations

An identity equation is a type of equation that is true for all possible values of the variable. When you simplify an equation and are left with a statement like \(-8 = -8\), as in our solved problem, you encounter an identity. This discovery means that no particular value for the variable is necessary to satisfy the equation since all values will work. Identity equations are often mistaken for having no solution initially, but recognizing them is essential:

• Recognition: If after simplification, the equation reduces to a true statement with no variables, it's an identity.
• Implication: It implies infinite solutions or that any value for the variable will satisfy the equation.

Understanding identity equations enriches our comprehension of algebraic solutions and highlights the importance of the simplification process in identifying unique or infinite solutions.