Problem 25 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(12 w+15 w-9+5=-3 w+5-9\)

Step-by-Step Solution

Answer

The solution is \(w = 0\). The equation is a standard linear equation with a unique solution.

Step 1: Combine like terms on the left side

First, combine the like terms on the left side of the equation. This involves adding the terms with the variable and the constant terms separately: \(12w + 15w - 9 + 5\)\(= 27w - 4\)

Step 2: Combine like terms on the right side

Next, combine the like terms on the right side of the equation. This involves adding the constants: \(-3w + 5 - 9\)\(= -3w - 4\)

Step 3: Set the simplified expressions equal to each other

Now set the simplified left side equal to the simplified right side: \(27w - 4 = -3w - 4\)

Step 4: Add 3w to both sides

To isolate the variable term, add \(3w\) to both sides: \(27w + 3w - 4 = -3w + 3w - 4\)\(30w - 4 = -4\)

Step 5: Add 4 to both sides

To isolate the variable, add \(4\) to both sides: \(30w - 4 + 4 = -4 + 4\)\(30w = 0\)

Step 6: Solve for w

Finally, solve for \(w\) by dividing both sides by 30: \(w = \frac{0}{30}\)\(w = 0\)

Step 7: Check the solution

Substitute \(w = 0\) back into the original equation to verify: \(12(0) + 15(0) - 9 + 5 = -3(0) + 5 - 9\)\(-4 = -4\) This confirms the solution is correct.

Step 8: Determine the type of equation

Since there is a valid solution, the equation is neither an identity nor a contradiction. It is a standard linear equation with a unique solution.

Key Concepts

Combining Like Terms

In algebra, combining like terms simplifies an equation, making it easier to solve. Like terms are terms that have the same variable raised to the same power. For example, in the equation: 12w + 15w - 9 + 5, you can combine 12w and 15w because both contain the same variable, w. Likewise, -9 and 5 are constant terms that can be combined. After combining these terms, the equation simplifies to: 27w - 4. Simplifying the equation by combining like terms is an essential first step in solving linear equations.

Isolating the Variable

Isolating the variable transforms an equation into a form where the variable stands alone on one side, facilitating its solution. For example, after simplifying the equation to 27w - 4 = -3w - 4, we add 3w to both sides to move the variable terms to the same side: 27w + 3w - 4 = -3w + 3w - 4. This simplifies to 30w - 4 = -4. The next step is to add 4 to both sides to isolate the variable w: 30w - 4 + 4 = -4 + 4 which simplifies to 30w = 0. Now, dividing both sides by 30 gives the value of the variable: w = 0.

Checking the Solution

After finding a solution, it's crucial to check your work by substituting the solution back into the original equation. This step ensures the solution is correct and satisfies the equation. For example, substituting w = 0 into the original equation 12w + 15w - 9 + 5 = -3w + 5 - 9: 12(0) + 15(0) - 9 + 5 = -3(0) + 5 - 9 simplifies to: -4 = -4. Since both sides are equal, the solution w = 0 is confirmed to be correct.

Unique Solution

A unique solution means the equation has exactly one solution. In our example, the equation 12w + 15w - 9 + 5 = -3w + 5 - 9 simplifies to 30w = 0, yielding the unique solution w = 0. This distinguishes it from identities (valid for any value of the variable) and contradictions (no value satisfies the equation). Recognizing that an equation has a unique solution is key to understanding its behavior and how it models real-world situations.