Problem 43 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$3 z=-2 z-15$$

Step-by-Step Solution

Answer

the solution for the equation is \(z = -3\)

Step 1: Simplify equation

Bring like terms together so that all terms containing \(z\) are on one side and the constants are on the other side.So, we will add \(2z\) to both sides:\(3 z + 2z = -2z + 2z -15which simplifies to:\(5z = -15\)

Step 2: Solve for the variable

Now we will isolate \(z\) by using the multiplication property of equality. We will divide each side of the equation by \(5\):\(5z \div 5 = -15 \div 5This gives us: \(z = -3\)

Step 3: Check the solution

To ensure that \(z = -3\) is the correct solution, we substitute it back into the original equation:\[3(-3) = -2(-3) - 15\]which simplifies to:\(-9 = 6 - 15\)and further simplifies to:\(-9 = -9\)This confirms that -3 is indeed the correct value of \(z\)

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a handy tool when solving linear equations. It allows us to add or subtract the same number from both sides of an equation without changing its equality. Think of it like balancing a scale. If you add or remove the same weight from each side, the scale remains balanced.

In our exercise, the equation was initially given as: \( 3z = -2z - 15 \). To get terms with \( z \) on one side, we added \( 2z \) to both sides. Doing so, we get:

• Left side: \( 3z + 2z \) which simplifies to \( 5z \)
• Right side: \( -2z + 2z - 15 \) which simplifies to \( -15 \)

Now, the equation has all \( z \)-terms on the left and constants on the right: \( 5z = -15 \). This step was crucial to organize the equation for solving the variable.

Multiplication Property of Equality

Once we've applied the Addition Property of Equality and organized our equation, the next step is often to use the Multiplication Property of Equality. This property states that you can multiply or divide both sides of an equation by the same non-zero number, and the equality will still hold.

In the given problem, from \( 5z = -15 \), we wanted to find the value of \( z \). To do this, we divided each side of the equation by \( 5 \), isolating \( z \) as:

• \( \frac{5z}{5} = \frac{-15}{5} \)
• Simplifying, we find that \( z = -3 \)

This step is pivotal in transitioning from a manipulated equation to a solved one, allowing us to pinpoint the exact value of \( z \). Remember, you must use a non-zero number; otherwise, the division would result in an undefined operation.

Checking Solutions

After solving an equation for a variable, it’s important to check your solution. This verification step ensures that your answer satisfies the original equation, eliminating possible errors along the way.

Using our solution of \( z = -3 \), we need to substitute it back into the original equation to confirm its correctness:

• Substitution: \( 3(-3) = -2(-3) - 15 \)
• Simplify each side: \(-9 = 6 - 15 \)
• Final result: \(-9 = -9 \)

Since both sides of the equation are equal, our solution \( z = -3 \) is confirmed as correct. This step is vital in identifying any calculation mistakes and ensures the integrity of your solution, building confidence in your ability to solve linear equations.