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. $$-25 b=0$$

Short answer

Question: Solve the following equations and check the solutions: 1) \(3x = -15\) 2) \(-25b = 0\) Answer: 1) \(x = -5\) 2) \(b = 0\)

Step-by-step solution

Identify the coefficient of the variable

In the first equation, \(3x = -15\), the coefficient of the variable \(x\) is \(3\).

Divide both sides by the coefficient

To isolate \(x\), divide both sides of the equation by \(3\): \(\frac{3x}{3} = \frac{-15}{3}\)

Simplify

After dividing, simplify the equation: \(x = -5\)

Check the solution

Plug the value of \(x\) back into the original equation to check the solution: \(3(-5) = -15\) \(-15 = -15\) The solution holds true, so \(x = -5\) is the correct answer. #Equation 2:#

Identify the coefficient of the variable

In the second equation, \(-25b = 0\), the coefficient of the variable \(b\) is \(-25\).

Divide both sides by the coefficient

To isolate \(b\), divide both sides of the equation by \(-25\): \(\frac{-25b}{-25} = \frac{0}{-25}\)

Simplify

After dividing, simplify the equation: \(b = 0\)

Check the solution

Plug the value of \(b\) back into the original equation to check the solution: \(-25(0) = 0\) \(0 = 0\) The solution holds true, so \(b = 0\) is the correct answer.

Key concepts

Coefficient

In algebra, the coefficient is a very important concept that refers to the number multiplied by the variable in an equation. Think of it as the ‘multiplier’ that stands in front of the variable. For example, in the equation \(3x = -15\), the number 3 is the coefficient of \(x\). This number tells us how many times \(x\) is being multiplied.

Understanding the role of the coefficient is essential because it directly affects the value of the variable. When solving linear equations, our main goal is often to find the value of these variables, and knowing the coefficients helps us manipulate the equations correctly. This leads us to 'isolating the variable', which will be our next step in solving the equation.

Isolate Variable

The term isolate the variable means to get the variable by itself on one side of the equation. Isolating the variable is like finding where the party is happening and then making sure that the variable is the only guest! For instance, in the equation \(3x = -15\), we want to get \(x\) alone on one side.

To do this, we have to perform the same operation on both sides of the equation. Since \(x\) is being multiplied by 3, we can cancel out this multiplication by dividing by 3. This crucial step will lead us right into understanding 'division in algebra', which allows us to isolate \(x\) effectively.

Division in Algebra

When it comes to division in algebra, think of it as the process of evening out pieces of a pie equally. It's the math way of sharing! We can use division to 'cancel out' the coefficient and isolate our variable.

How to Share the Pie Equally

In our equation \(3x = -15\), by dividing both sides by 3, we are essentially sharing both sides of the equation into three equal parts.

When we divide \(3x\) by 3, we are left with just \(x\), because \(3/3 = 1\), and \(1 \cdot x = x\). Similarly, when we divide -15 by 3, we get -5. So the division helped us to isolate \(x\) and find that \(x = -5\). Always remember to perform the same division on both sides to maintain the balance of the equation. Let's see how we can check if our solution is correct in the next section.

Solution Checking

Ensuring that our solution is correct is like checking our work after completing a craft project—it's essential to confirm we've done everything right. Solution checking in algebra involves substituting the found value of the variable back into the original equation.

If we rewind back to our equation \(3x = -15\) and replace the \(x\) with -5, as we found that \(x = -5\), we should get the original equation to balance. This is exactly what happens because \(3(-5) = -15\) indeed simplifies to \( -15 = -15\). The same process applies to checking \(b = 0\) in the equation \( -25b = 0\).

By substituting 0 for \(b\), we confirm the solution as the equation balances out to \(0 = 0\). Through solution checking, we gain confidence that the solutions we have determined are correct, allowing us to proceed with assurance.