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. $$-12 t=48$$

Short answer

The value of x is -5, and the value of t is -4.

Step-by-step solution

Solve for x in the first equation

In the first equation, we have \(3x=-15\). To solve for \(x\), we will divide both sides of the equation by 3: \(x = \frac{-15}{3}\)

Calculate the value of x

Now, let's divide -15 by 3 to find the value of \(x\): \(x = -5\)

Check the solution for the first equation

In order to check if our solution is correct, let's plug the value of \(x\) back into the first equation: \(3(-5) = -15\) Evaluating the left side of the equation, we get: \(-15=-15\) Since both sides of the equation are equal, our solution for \(x\) is correct.

Solve for t in the second equation

In the second equation, we have \(-12t=48\). To solve for \(t\), we will divide both sides of the equation by -12: \(t = \frac{48}{-12}\)

Calculate the value of t

Now, let's divide 48 by -12 to find the value of \(t\): \(t = -4\)

Check the solution for the second equation

In order to check if our solution is correct, let's plug the value of \(t\) back into the second equation: \(-12(-4) = 48\) Evaluating the left side of the equation, we get: \(48=48\) Since both sides of the equation are equal, our solution for \(t\) is correct.

Final Solution

So the solutions to the given equations are: \(x = -5\) for the equation \(3x = -15\) and \(t = -4\) for the equation \(-12t = 48\).

Key concepts

Division in Equations

When solving linear equations, one of the fundamental steps is to isolate the variable. A key method used is division. If you ever encounter an equation where the variable is being multiplied by a number, like in the case of the equation \(3x = -15\), the goal is to undo this multiplication. You do this by dividing both sides of the equation by the number that the variable is being multiplied with.

For our equation, you would divide both sides by 3, leading to:

• Left Side: \(\frac{3x}{3} = x\)
• Right Side: \(\frac{-15}{3} = -5\)

This simplifies to \(x = -5\). By dividing, you essentially neutralize the multiplication and leave the variable by itself, giving you the solution. This method is critical as it ensures that both sides of the equation remain balanced.

Checking Solutions

After solving an equation, it is always a good practice to double-check your solution. This means substituting the found value back into the original equation to see if it holds true. Let's use the example we solved earlier.

• Original Equation: \(3x = -15\)
• Solution: \(x = -5\)

Now substitute \(-5\) back into the equation:

• \(3(-5) = -15\)

Evaluating this, \(-15 = -15\). The equality confirms that \(x = -5\) is indeed the correct solution.

This confirmation step is crucial as it verifies the accuracy of your calculation and ensures you didn't make any errors along the way.

Variable Isolation

Variable isolation is a technique used to get the variable 'alone' on one side of the equation. The goal is to manipulate the equation so that the variable is on one side and a constant is on the other. This often involves performing inverse operations like addition, subtraction, multiplication, or division.

Consider the equation \(-12t = 48\). To isolate \(t\), we need to eliminate the \(-12\) from the left side. Since \(t\) is multiplied by \(-12\), divide both sides by \(-12\) to isolate \(t\):

• Left Side: \(\frac{-12t}{-12} = t\)
• Right Side: \(\frac{48}{-12} = -4\)

This results in \(t = -4\). Thus, variable isolation helps you to find the precise value of the variable by getting it by itself.

Mathematical Operations

In solving equations, mathematical operations like addition, subtraction, multiplication, and division go hand in hand. Understanding when and how to perform these operations is essential. The order of these operations ensures a correct and clear simplification process.

For division in our context, it reverses multiplication, which is seen in equations like \(3x = -15\). When we divided \(-15\) by \(3\), we simplified the equation using basic arithmetic:

• \(-15 \div 3 = -5\)

You must consistently apply these operations to each side of an equation to maintain balance. Errors typically occur when steps are skipped or operations are misapplied. Always follow a logical sequence and double-check work to ensure equations are solved correctly.