Problem 35 - 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. $$ \frac{-6 m}{5}+11=-13 $$

Step-by-Step Solution

Answer

Answer: The given equation is a conditional equation, and the value of m for which the equation is true is m = 20.

Step 1: Get rid of any constants on the same side as the variable

To begin solving for \(m\), let's first move any constants on the same side as the variable to the other side of the equation for a simplified equation. In this case, we want to move the \(+11\) to the other side of the equation. Currently, we have: $$ \frac{-6 m}{5} + 11 = -13 $$ Subtract \(11\) from both sides of the equation: $$ \frac{-6 m}{5} = -24 $$

Step 2: Get rid of the fraction

To remove the fraction and have a simpler equation, let's multiply both sides of the equation by the denominator of the fraction, which is \(5\) in our case. Multiply both sides by \(5\): $$ 5 \cdot \frac{-6 m}{5} = -24 \cdot 5 $$ This simplifies to: $$ -6 m = -120 $$

Step 3: Solve for m

Finally, we only need to isolate \(m\) by dividing both sides of the equation by \(-6\): $$ \frac{-6 m}{-6} = \frac{-120}{-6} $$ This simplifies to: $$ m = 20 $$ Since we found a unique solution for the variable, the given equation is a conditional equation. And the solution is \(m = 20\).

Key Concepts

Solving Equations

Understanding how to solve equations is a fundamental skill in algebra and critical for tackling more advanced math topics. When faced with an equation, the primary goal is to isolate the variable in question on one side of the equation. This allows us to find its value. In our original exercise, we started with:\[\frac{-6 m}{5} + 11 = -13\]First, we aimed to remove the constant, \(11\), by subtracting it from both sides, which simplified the equation. It's essential to apply the same operation to both sides to maintain the equation's balance:

• Remember, in solving equations, whatever you do to one side, you must do to the other.
• This keeps the equation balanced, ensuring that you don't alter the equality.
• The goal is to eventually have the variable alone.

After simplifying, you'll often move on to dealing with fractions or coefficients that are attached to the variable, as discussed in our next section.

Fractions in Algebra

Dealing with fractions can be tricky, but they are common in algebra. In the equation given: \[\frac{-6 m}{5} + 11 = -13\]fraction \(\frac{-6 m}{5}\) appears, which can seem daunting, but the strategy is to eliminate the fraction early in your calculations. We achieved this by multiplying both sides by the denominator (5), simplifying our equation to \[-6 m = -120\]. Doing this makes the equation easier to manage without the complication of fractions:

• Identify the denominator of the fraction to guide what you multiply both sides of the equation by.
• Multiplying by the denominator cancels the fraction, reducing complexity.

Thus, simplifying makes it seamless to move to the next step where you isolate the variable.

Linear Equations

A linear equation is an equation of the first degree, meaning its highest exponent of the variable is one. These equations are straightforward once you grasp their fundamental structure. Take our equation after eliminating the fraction:\[-6 m = -120\]This setup is the classic form of a linear equation, where you only need to do one more thing: isolate the variable \(m\).To solve, divide both sides by the coefficient of the variable (-6), yielding:\[m = 20\]. The solution derived represents a unique solution typical for conditional equations, as seen here:

• Linear equations generally appear as a straight line when graphed, parallel with typical solutions like the one seen.
• Remember to check if there is only one variable to ensure it remains linear.
• Efficiently learned steps help simplify and solve linear equations in various conditions."