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. $$-18 m=0$$

Short answer

This simplifies to: $$ x = -5 $$ So the value of x in the equation \(3x = -15\) is -5. To check our solution, we can substitute -5 back into the original equation: $$ 3(-5) = -15 $$ $$ -15 = -15 $$ Since both sides of the equation are equal, our solution is correct. The value of x is -5.

Step-by-step solution

Find the value of x in the equation \(3x = -15\)

To isolate the variable x, we need to divide both sides of the equation by 3: $$ \frac{3x}{3} = \frac{-15}{3} $$

Key concepts

Isolate the Variable

When solving linear equations, one of the most crucial steps is to isolate the variable. What this means is that you want to have the variable by itself on one side of the equation. To achieve this, you will often perform operations that 'undo' whatever is being done to the variable.
To isolate the variable in an equation like \(3x = -15\), you would look at what is happening to \(x\). Here, \(x\) is being multiplied by 3. To undo this multiplication, you would perform the opposite operation, which is division in this case.

• Identify what operation is being performed on the variable.
• Use the inverse operation to undo the effect.
• Apply this inverse operation to both sides of the equation to keep it balanced.

This balanced approach ensures that whatever changes you make to one side of the equation, you also make to the other, thus maintaining equality.

Division Operation

In arithmetic and algebra, the division operation is essentially the inverse of multiplication. When you divide, you are splitting a number into equal parts or groups.
The division operation is symbolized with a slash \(/\), or a horizontal line, which in more complex mathematics may also be referred to as a fraction line. For example, when you see \(\frac{3x}{3}\), it's asking how many times does 3 go into \(3x\)?

• Division undoes multiplication; if a variable is multiplied by a number, dividing by the same number will 'free' the variable.
• Remember that division by zero is not permitted, as it does not have a meaning within the real numbers.

The result of this operation provides the value for \(x\) when solving an equation like the given one, \(3x = -15\).

Inverse Operations

Inverse operations in mathematics are pairs of operations that undo each other. The addition and subtraction operations are inverses, as are multiplication and division.
Understanding inverse operations is key to solving equations because they allow you to simplify equations and isolate variables. For the equation \(3x = -15\), since \(x\) is multiplied by 3, the inverse operation to isolate \(x\) is division by 3.

• Use subtraction, if the variable has been added to another number.
• Use addition, if the variable has been subtracted from another number.
• Use multiplication, if the variable has been divided by a number.
• Use division, if the variable has been multiplied by a number.

After applying inverse operations, it’s essential to verify that both sides of the equation still have the same value, ensuring that the initial equation remains true.

Equation Solution Checking

Once an equation has been solved, it's vital to check the solution to ensure it is correct. To check the solution of an equation, like the value obtained for \(x\) in \(3x = -15\), you substitute your answer back into the original equation.
If both sides of the equation are equal after the substitution, it confirms that the solution is correct.

• Substitute the solution into the original equation.
• Simplify both sides of the equation.
• If both sides are equal, the solution is correct.
• If they're not equal, re-examine the steps to find the error.

This process not only verifies the solution but also reinforces your understanding of the relationship between the operations performed and the equation’s balance.