Problem 44 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$2 z=-4 z+18$$

Step-by-Step Solution

Answer

The solution to the equation \(2z = -4z + 18\) is \(z = 3\), this was confirmed by substituting \(z = 3\) back into the original equation and observing that both sides are indeed equal.

Step 1: Group Terms with z

In the first step, all the z terms need to be brought to one side of the equation in order to simplify the equation. This can be done using the addition property of equality by adding \(4z\) to both sides. So the equation becomes: \(2z + 4z = 18\)

Step 2: Simplify Equation

Next, the equation can be simplified by combining like terms on the left side, resulting in: \(6z = 18\)

Step 3: Solve for z

Then use the multiplication property of equality to solve for \(z\). To isolate \(z\), divide both sides of the equation by 6. This gives: \(z = 18 / 6\)

Step 4: Check the Solution

Finally, to check if the solution is correct, substitute the value of \(z\) back into the original equation: \(2(3) = -4(3) + 18\). On simplifying the left and right-hand side, both sides equal 6, hence confirming the solution is correct.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental principle used in solving equations. It states that if you add the same number to both sides of an equation, the equality of the equation remains unchanged. This means that the equation stays balanced, and you haven’t changed the equality’s truth.

In our problem, we started with the equation:

• \(2z = -4z + 18\)

To group the terms with \(z\) on one side, we added \(4z\) to both sides of the equation. By carrying out the same addition on each side, the equation remained balanced:

• \(2z + 4z = 18\)

This application of the addition property allowed us to combine all the \(z\) terms, simplifying the equation into a more workable form.

Multiplication Property of Equality

The Multiplication Property of Equality is another crucial tool when solving equations. This property suggests that if you multiply or divide both sides of an equation by the same nonzero number, the equality will still hold true. It's essential when you need to isolate a variable.

In the given equation, following the application of the addition property, we simplified it to:

• \(6z = 18\)

To solve for \(z\), we used division, a form of multiplication in terms of equality, by dividing both sides by 6:

• \(z = \frac{18}{6}\)
• \(z = 3\)

By doing this, we successfully isolated \(z\), obtaining the solution. This step illustrates how multiplication and division can be used to ensure that the equation remains balanced while isolating the variable.

Checking Solutions

Checking solutions is an essential step to verify the correctness of an answer while solving equations. Once you obtain a solution, inserting it back into the original equation helps confirm the equality remains valid.

For our problem, the found solution was \(z = 3\). By substituting \(z = 3\) back into the original equation:

• \(2(3) = -4(3) + 18\)

Calculating both sides gives us:

• Left side: \(6\)
• Right side: \(-12 + 18 = 6\)

Since both sides are equal, we have confirmed that our solution, \(z = 3\), is indeed correct. This step is crucial in ensuring that no errors were made during earlier calculations.