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. $$5 x=-125$$

Short answer

Answer: The solutions are \(x = -5\) for the first equation and \(x = -25\) for the second equation. Both solutions are correct as they satisfy their respective original equations.

Step-by-step solution

Solve the first equation: \(3x = -15\)

To isolate and find the value of \(x\), we will divide both sides of the equation by \(3\): $$\frac{3x}{3} = \frac{-15}{3}$$ This simplifies to: $$x = -5$$

Check the solution for the first equation

To check our solution, plug the value of \(x\) back into the original equation: $$3(-5) = -15$$ Multiply it out: $$-15 = -15$$ Since both sides are equal, our solution is correct.

Solve the second equation: \(5x = -125\)

To isolate and find the value of \(x\), we will divide both sides of the equation by \(5\): $$\frac{5x}{5} = \frac{-125}{5}$$ This simplifies to: $$x = -25$$

Check the solution for the second equation

To check our solution, plug the value of \(x\) back into the original equation: $$5(-25) = -125$$ Multiply it out: $$-125 = -125$$ Since both sides are equal, our solution is correct. To summarize, the solutions for each equation are \(x = -5\) for the first equation and \(x = -25\) for the second equation.

Key concepts

Division in Equations

When solving linear equations, one of the most common techniques you will encounter is division. Division helps to isolate the variable you are trying to solve for. When you have an equation like \(3x = -15\), you need to turn that equation into something simpler: \(x = \).
First, look at the term with the variable, which in this case is \(3x\). The goal is to get \(x\) by itself on one side of the equation. To do this, you divide both sides of the equation by the same number, which is the coefficient of \(x\) (in this case, \(3\)).

• Example: Divide \(3x\) and \(-15\) by \(3\) resulting in \(x = -5\).

By dividing, you maintain balance, ensuring what you do to one side of the equation, you also do to the other. This fundamental property of equations is what keeps them valid while manipulating them to find solutions.

Checking Solutions

Once you arrive at a potential solution for an equation, it’s important to check if your solution is correct. Checking solutions ensures your calculations are accurate and you've understood the process correctly.
To check, substitute the solution back into the original equation. For instance, if your solution for \(x\) is \(-5\) in the equation \(3x = -15\), you replace \(x\) with \(-5\):

• Substitution: \(3(-5)\)
• Calculate: \(-15 = -15\)

If both sides of the equation remain equal after the substitution, then your solution is correct. This simple step prevents mistakes and reinforces the learning process by verifying your answer.

Prealgebra Concepts

Understanding prealgebra concepts is crucial when starting with equations. These concepts form the foundation of algebra, making calculations and problem-solving much easier.
Prealgebra includes concepts like:

• The order of operations (PEMDAS/BODMAS).
• Basic arithmetic operations: addition, subtraction, multiplication, and division.
• Understanding variables and constants in equations.
• Balancing equations by performing the same operation on both sides.

In the context of our equations, knowing how to properly manipulate and simplify equations is key. For instance, dividing both sides of \(5x = -125\) by \(5\) results in \(x = -25\). Prealgebra teaches you to see that each equation step logically follows from the previous step, aiding both understanding and retention.