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. $$9 x=-18$$

Short answer

Answer: The solutions for the given equations are \(x = -5\) and \(x = -2\).

Step-by-step solution

Solve the first equation

Divide both sides of the equation \(3x = -15\) by 3 to isolate x: \(x = \frac{-15}{3}\) Now, simplify the fraction: \(x = -5\)

Check the solution for the first equation

Substitute the value of x into the original equation: \(3(-5) = -15\) Simplify the expression: \(-15 = -15\) Since both sides of the equation are equal, the solution \(x = -5\) is correct.

Solve the second equation

Divide both sides of the equation \(9x = -18\) by 9 to isolate x: \(x = \frac{-18}{9}\) Now, simplify the fraction: \(x = -2\)

Check the solution for the second equation

Substitute the value of x into the original equation: \(9(-2) = -18\) Simplify the expression: \(-18 = -18\) Since both sides of the equation are equal, the solution \(x = -2\) is correct. In conclusion, the solutions for the given equations are \(x = -5\) and \(x = -2\).

Key concepts

Solving Equations

Solving equations with variables is like finding the missing piece of a puzzle. You are given an equation with a variable, such as \(3x = -15\), and your goal is to find the value of \(x\) that makes the equation true. This process often involves isolating the variable on one side of the equation. For example, in the equation \(3x = -15\), you can solve for \(x\) by dividing both sides of the equation by 3. This gives \(x = \frac{-15}{3}\), which simplifies to \(x = -5\). By isolating \(x\), you can see clearly what value \(x\) needs to be to make the equation true.

Division in Algebra

The process of division in algebra is crucial for isolating variables in equations. When you have an equation like \(9x = -18\), you perform division to make the equation simpler and find \(x\). Here, you divide both sides by 9 to get \(x = \frac{-18}{9}\), simplifying to \(x = -2\).

Division helps break down more complex expressions, making calculations easier and more straightforward. By ensuring you perform the same operation on both sides of an equation, you maintain balance, keeping the equation true. When dividing:

• Identify the number you need to divide by from both sides.
• Perform the division on each side.
• Double-check that your operations keep the equation balanced.

This assures that your solution is reached appropriately.

Checking Solutions

Once you've found a solution to an equation, it is always important to check your work. This step ensures that you did not make any mistakes while solving the equation. To check the solution, substitute the value of the variable back into the original equation and verify that both sides are equal.

For instance, if your solution for \(3x = -15\) was \(x = -5\), you would substitute \(-5\) into the equation: \(3(-5) = -15\). Simplify the expression, and if both sides are equal, like \(-15 = -15\), then your solution is correct.

• Substitute the found value back into the original equation.
• Perform the arithmetic to simplify both sides of the equation.
• Ensure both sides of the equation hold equal values.

By checking solutions, you validate your approach and confidently confirm that your answer is accurate.