Problem 8 - Prealgebra

Question

Solve each equation. Check your solution. $$12-h=-h+3$$

Step-by-Step Solution

Answer

There is no solution for the equation.

Step 1: Combine Like Terms

Start by moving all the terms involving \(h\) to one side of the equation. Add \(h\) to both sides of the equation: \[12 - h + h = -h + h + 3\]This simplifies to:\[12 = 3\]

Step 2: Simplify

Notice that after combining like terms, the variable \(h\) has been canceled out, resulting in a false equality statement \(12 = 3\).

Step 3: Interpret the Result

Since we have arrived at an untrue statement, this means that there is no solution for the given equation. The original setup results in a contradiction, so there is no value of \(h\) that satisfies the equation.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in prealgebra and involves finding the value of an unknown variable. The goal is to isolate the unknown variable on one side of the equation to determine its value.
To solve an equation:

• Start by simplifying both sides of the equation if necessary.
• Move all terms with the variable to one side of the equation.
• Use inverse operations to isolate the variable. For instance, if a term is added to the variable, subtract it from both sides.
• Finally, check your solution by substituting it back into the original equation to ensure it holds true.

In prealgebra, these steps help efficiently find solutions to various equations, but it's important to understand that some equations may not follow this standard path.

No Solution

It might come as a surprise that some equations do not have a solution at all. This happens when we end up with a false statement after attempting to solve the equation.
For example, you may arrive at something like 12 = 3, which is clearly not true. In such cases, we determine that there's no solution because no value will satisfy the original equation.

• If combining terms leads to contradictory statements, it indicates the equation has no solution.
• An equation with no solution is sometimes referred to as "inconsistent."

It's crucial to recognize this result, as it reflects that the equation is set up in a way that makes finding a solution impossible.

Combining Like Terms

Combining like terms is a simplifying process used in algebra to make equations easier to solve. Like terms are terms that contain the exact same variables raised to the same power.
For instance, in the equation \(12 - h = -h + 3\), both \(-h\) terms are like terms. To combine them:

• Add \(h\) to both sides, which effectively cancels out \(-h\) on the left side.
• The right side \(-h + h + 3\) becomes \(3\) after simplifying.

This process helps in reducing the number of terms in an equation, providing a clearer path toward finding a solution or, as seen here, realizing that there is no solution.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to solve equations or reorder them. It's essential in dealing with equations that might not immediately give up their solutions.
Common manipulations include:

• Adding, subtracting, multiplying, or dividing both sides of an equation by the same number.
• Rearranging terms to get the variable of interest on one side of the equation.
• Cancelling out terms, as done by adding \(h\) to both sides in our original equation.

Algebraic manipulation helps to systematically simplify equations, revealing contradictions or solutions, depending on the given equation's structure.