Problem 14 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$3(3 z+5)-7=89$$

Step-by-Step Solution

Answer

The solution is \(z = 9\)

Step 1: Distribute or Expand the Bracket

First, distribute the 3 to both the terms inside the bracket: \(3 \cdot 3z + 3 \cdot 5 - 7 = 89\). This simplifies to \(9z + 15 - 7 = 89\).

Step 2: Simplify the Equation

Combine like terms on the left side of the equation to simplify it: \(9z + 8 = 89\).

Step 3: Isolate the Variable Term

To isolate the variable term, subtract 8 from both sides of the equation: \(9z = 81\).

Step 4: Calculate the Solution

Finally, divide both sides of the equation by 9 to solve for z: \(z = 9\).

Step 5: Check the Solution

Substitute z=9 back into the original equation to see if it holds true. \(3(3 \cdot 9 + 5) - 7 = 89\), which simplifies to \(3(27 + 5) - 7= 89\), and further to \(3 \cdot 32 - 7 = 89\), and finally to \(89 = 89\), which is true, therefore the solution z=9 is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps in simplifying equations, especially those that involve multiplication over addition or subtraction within parentheses. When you see an expression like \(3(3z + 5)\), this indicates that you need to multiply the number outside the parentheses, which is 3 in this case, by each term within the parentheses.
To apply the distributive property properly, follow these steps:

• Multiply the number outside the bracket with each term inside. For our example, you multiply 3 with \(3z\) to get \(9z\).
• Then, multiply 3 with 5 to get 15.

The result from applying the distributive property to \(3(3z + 5)\) is \(9z + 15\).
This property is crucial because it allows us to eliminate the parentheses and transform a complex expression into a simpler one, thereby making it easier to solve.

Isolating Variables

Isolating the variable is a key step in solving linear equations. The goal here is to get the variable by itself on one side of the equation. This process allows us to find the value of the variable that makes the equation true. Let's look at the equation you get after using the distributive property: \(9z + 15 - 7 = 89\).
Here's how you can isolate the variable \(z\):

• First, combine like terms to simplify. Here, subtract 7 from 15 to result in 8, leading to the equation \(9z + 8 = 89\).
• Next, you want to shift the constant term on the same side as the variable (which is 8 in this equation) to the other side. Subtract 8 from both sides to keep the equation balanced, transforming it to \(9z = 81\).
• Finally, to isolate \(z\) completely, divide both sides by 9, which yields \(z = 9\).

After isolating the variable, you have the solution \(z = 9\). This signifies the value of \(z\) that solves the equation.

Checking Solutions

Once you find a solution, it’s important to verify it to ensure it satisfies the original equation. This step helps confirm that no mistakes were made during the calculations.
To check the solution of \(z=9\) for our original equation \(3(3z + 5) - 7 = 89\):

• Substitute 9 back into the original equation: \(3(3 \cdot 9 + 5) - 7\).
• Calculate inside the parentheses first: \(3 \cdot 9 + 5\) gives you \(32\).
• Next, multiply 3 by 32 to get \(96\).
• Subtract 7 from 96, resulting in \(89\).
• If both sides of the equation are equal, as they are in this case since 89 equals 89, then the solution is correct.

Checking solutions is important as it validates your work by ensuring that the variable value does indeed satisfy the original equation.