Problem 43 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ 12 n+5=5 n-16 $$

Step-by-Step Solution

Answer

Question: Solve the conditional equation: \(12n + 5 = 5n - 16\). Answer: \(n = -3\).

Step 1: Subtract 5n from both sides

To get all the n terms on one side, we'll subtract 5n from both sides of the equation: $$ 12n + 5 - 5n = 5n - 16 - 5n $$ This simplifies to: $$ 7n + 5 = -16 $$

Step 2: Subtract 5 from both sides

Now we'll isolate the n term by subtracting 5 from both sides: $$ 7n + 5 - 5 = -16 - 5 $$ This simplifies further to: $$ 7n = -21 $$

Step 3: Divide by 7

Finally, we'll divide both sides by 7 to find the value of n: $$ \frac{7n}{7} = \frac{-21}{7} $$ This simplifies to: $$ n = -3 $$ So our solution for the given conditional equation is \(n=-3\).

Key Concepts

Conditional Equation

A conditional equation is one that is true for certain values of the variables involved. In simple words, it only holds under specific conditions or constraints. For example, the equation in our exercise, \(12n + 5 = 5n - 16\), is only true for a particular value of \(n\). If you solve it properly, you'll find that \(n = -3\).

• Conditional equations can have solutions like numbers or expressions.
• They are different from other types of equations, such as identities, which are always true.
• They are also not contradictions, which are never true for any variable values.

A good way to spot a conditional equation is to attempt solving it. If you're able to find a specific solution, you've likely got a conditional equation on your hands.

Subtraction Method

The subtraction method in mathematics is a technique to simplify equations by removing terms from both sides. In our example, the subtraction method helps in gathering like terms together. We used this method to set all the \(n\) terms on one side of the equation.

Here's how it works:

• Subtract the same expression from both sides of the equation. In our case, we subtracted \(5n\) from both sides.
• This step creates a simpler equation to work with: \(7n + 5 = -16\).
• Doing so helps in getting closer to isolating the variable we are solving for.

It's a simple yet effective means of simplifying complex equations and is one of the foundational techniques in algebra. By consistently applying the subtraction method, solving equations becomes much more manageable.

Isolation of Variable

The isolation of the variable is an essential step in solving equations. Put simply, it means getting the variable by itself on one side of the equation. This allows you to easily determine the variable's value.

• In the worked example, after subtracting terms, we got \(7n\) by itself, making it easier to handle.
• Isolation often involves simplifying terms and removing numbers or coefficients from one side of the equation.
• Remember, whatever you do to one side of the equation, you must do to the other side, preserving the balance.

After isolating the variable, you can proceed with any additional steps necessary to fully solve the equation, such as dividing or multiplying.

Division Method

The division method is a crucial step when you're left with a simple expression like \(7n = -21\). Here, the goal is to completely solve for the variable by making its coefficient one. This involves dividing both sides of the equation by the number that is currently multiplying the variable.

• In our problem, after isolating \(n\) by getting rid of other numbers, we divided both sides by \(7\).
• This simplified the equation to \(n = -3\), giving us the final solution.
• Dividing by the coefficient ensures the variable \(n\) doesn't have anything attached to it, making it precisely what you solve for.

Division method is often one of the final steps in solving an equation. It's simple: just divide every term by the coefficient of the variable to discover its exact value. This technique is straightforward but pivotal in equation-solving.