Question

Prep Exercise 8 For the following equation, complete the related division equation used to solve the equation. \(3 x=-15\) \(x=\) ____ solve and check. $$-6 m=-54$$

Short answer

Answer: The solutions to the given equations are: \(x = -5\) and \(m = 9\).

Step-by-step solution

Solve the first equation for x

To solve the equation \(3x = -15\), we need to isolate x. We can do this by dividing both sides of the equation by 3: \(x = \frac{-15}{3}\) Now, we can simplify the right side of the equation to find the value of x: \(x = -5\)

Check the first equation

To check if the solution \(x=-5\) is correct, we can substitute the value of x back into the original equation: \(3(-5) = -15\) This simplifies to: \(-15 = -15\) Since the equation is true, we can confirm that \(x=-5\) is the correct solution.

Solve the second equation for m

To solve the equation \(-6m = -54\), we need to isolate m. We can do this by dividing both sides of the equation by -6: \(m = \frac{-54}{-6}\) Now, we can simplify the right side of the equation to find the value of m: \(m = 9\)

Check the second equation

To check if the solution \(m=9\) is correct, we can substitute the value of m back into the original equation: \(-6(9) = -54\) This simplifies to: \(-54 = -54\) Since the equation is true, we can confirm that \(m=9\) is the correct solution.

Key concepts

Division Property of Equality

The division property of equality is a fundamental concept used in algebra to solve equations. This principle states that if you divide both sides of an equation by the same non-zero number, the two sides remain equal. In other words, the balance of the equation is preserved.
This property is particularly useful when you need to isolate a variable. By dividing both sides of the equation by the coefficient of the variable, we can solve for the variable.

• If you have an equation in the form of \(ax = b\), you can isolate \(x\) by dividing both sides by \(a\): \(x = \frac{b}{a}\).
• In the exercise, for the equation \(3x = -15\), dividing both sides by \(3\) isolates \(x\): \(x = \frac{-15}{3} = -5\).
• Similarly, for \(-6m = -54\), dividing both sides by \(-6\) isolates \(m\): \(m = \frac{-54}{-6} = 9\).

Understanding this concept is crucial for solving linear equations efficiently.

Substitution Method

Once you solve an equation and find the value of a variable, it's important to ensure the solution is correct. This is where substitution comes in handy.
By substituting the solution back into the original equation, you can verify whether it satisfies the equation.
Let's see how this works with our practice problems:

• For the equation \(3x = -15\), after finding \(x = -5\), substitute \(-5\) into the equation: \(3(-5) = -15\). This checks out, as it simplifies to \(-15 = -15\), confirming the solution is correct.
• For the second equation \(-6m = -54\), after finding \(m = 9\), substitute \(9\) into the equation: \(-6(9) = -54\). This also checks out, simplifying to \(-54 = -54\).

This method ensures that the solutions found are not just mathematically calculated but also accurate within the context of the original problem.

Checking Solutions

Checking your solution is like giving your work a final coat of polish to ensure accuracy. It involves verifying that your calculated solution satisfies the original equation. This step helps prevent errors and guarantees the solution truly solves the problem.
Here’s why and how you should check your solutions:

• Ensures accuracy: By substituting the solution back into the equation, you can confirm that both sides are equal, which verifies the solution’s correctness.
• Identifies mistakes: If the two sides do not equal each other after substitution, it indicates a potential error either in calculation or in understanding the problem premise.
• In our examples: For \(3x = -15\), substituting \(x = -5\) resulted in \(3(-5) = -15\), and for \(-6m = -54\), substituting \(m = 9\) resulted in \(-6(9) = -54\). Both checks confirmed correct solutions.

This step is crucial in reinforcing confidence in your solution and ensuring that all calculations have been done correctly.