Problem 24 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$24=12(z+1)-3(4 z-2)$$

Step-by-Step Solution

Answer

The equation has no solution because the simplified equation, 24 = 18, is not correct.

Step 1: Expand Brackets

Firstly, apply the distributive property (a(b + c) = ab + ac), the equation becomes 24 = 12z + 12 - 12z + 6.

Step 2: Simplify and Combine Like Terms

Now, combine the z-terms and the constant terms on the right side of the equation. That gives 24 = 18.

Step 3: Comparison and Solution

We can now observe that the simplified equation states 24 = 18, which in fact is not correct. Therefore, the original equation has no solution because there's no value for z that would make this equation true.

Key Concepts

Distributive Property

The distributive property is a fundamental part of algebra, allowing you to simplify equations and make them easier to solve. When you see an expression like \(a(b + c)\), you can expand it to \(ab + ac\). This rule helps in spreading or distributing one term over the others within parentheses. Here it was applied to \(12(z+1)\) and \(-3(4z - 2)\):

• \(12(z+1)\) turns into \(12z + 12\).
• \(-3(4z-2)\) becomes \(-12z + 6\).

By rewriting expressions like this, you can more easily identify the terms that need to be combined or canceled out in order to solve the equation. In this exercise, distributing simplified the equation by breaking it down into separate components.

Like Terms

Combining like terms is another essential technique in algebra, crucial for simplifying equations. Like terms are terms whose variables (and their exponents) are the same. Simplifying these makes equations much tidier and easier to solve. When you expand an expression, you often find similar terms that can be combined. For example:

• After distributing, the equation was \(24 = 12z + 12 - 12z + 6\).
• Here, \(12z\) and \(-12z\) are like terms because they both contain \(z\).
• The constant terms \(12\) and \(6\) can also be combined to give \(18\).

Combining like terms here simplifies the equation to \(24 = 18\). Simplification like this not only tidies up your work but also reveals the nature of your equation more clearly, as seen with this unsolvable one.

No Solution Equations

An equation is said to have "no solution" when, after simplifying, you encounter a false statement. This means there is no possible value for the variable that makes the equation true. In this particular exercise, after simplifying the equation using the distributive property and combining like terms, we ended up with \(24 = 18\). Clearly, this statement is false, indicating there's no solution.

Equations can typically end up with three types of scenarios:

• One solution: The equation simplifies to a true statement like \(x = 5\).
• No solution: It simplifies to a false statement as in our case here, \(24 eq 18\).
• Infinite solutions: Typically simplifies to an identity like \(0 = 0\), applicable in cases where any value is a solution.

Recognizing that an equation cannot be solved is just as important as solving it, showing comprehension of algebraic principles rather than just the mechanics.