Problem 28 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ 0.60(z-300)+0.05 z=0.70 z-205 $$

Step-by-Step Solution

Answer

The solution is \( z = 500 \).

Step 1: Distribute the Coefficient

Start by distributing the 0.60 to the terms inside the parentheses: \[ 0.60(z - 300) = 0.60z - 0.60 imes 300 \] This simplifies to: \[ 0.60z - 180 \] So, the original equation becomes: \[ 0.60z - 180 + 0.05z = 0.70z - 205 \]

Step 2: Combine Like Terms on One Side

Combine the like terms on the left side of the equation. You have \(0.60z + 0.05z\), which sums up to \(0.65z\). Now the equation is: \[ 0.65z - 180 = 0.70z - 205 \]

Step 3: Isolate the Variable Term

To isolate the variable \(z\), move the \(0.70z\) term to the left side by subtracting \(0.70z\) from both sides: \[ 0.65z - 0.70z - 180 = -205 \] Simplifying the \(z\)-terms gives: \[ -0.05z - 180 = -205 \]

Step 4: Solve for the Variable

Next, isolate \(-0.05z\) by adding 180 to both sides of the equation: \[ -0.05z = -205 + 180 \] Simplify the right side: \[ -0.05z = -25 \] Now, solve for \(z\) by dividing both sides by \(-0.05\): \[ z = \frac{-25}{-0.05} \] Simplifying gives: \[ z = 500 \]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to remove parentheses by distributing a multiplied value over each term inside the parentheses.
This property is especially useful when solving linear equations, as it helps to simplify expressions and equations. In our example, we apply the distributive property to the term \(0.60(z - 300)\), which involves multiplying the coefficient \(0.60\) with each term inside the parentheses.

• First, multiply \(0.60\) by \(z\), giving \(0.60z\).
• Next, multiply \(0.60\) by \(300\), resulting in \(180\).

So, the expression \(0.60(z - 300)\) becomes \(0.60z - 180\).
This allows us to eliminate the parentheses and proceed to the next steps of solving the equation.

Combining Like Terms

Combining like terms is an important step in equation simplification. It involves adding or subtracting terms that have the same variable or are constants. This helps to condense the equation, making it easier to manage.
In the example equation, after applying the distributive property, we have \(0.60z - 180 + 0.05z = 0.70z - 205\).

• Here, \(0.60z\) and \(0.05z\) are like terms because both are multiplied by\(z\). Add them together to get \(0.65z\).
• The constants \(-180\) and \(-205\) are not combined at this step since they are on different sides of the equation.

Once like terms are combined, the equation becomes simpler, \(0.65z - 180 = 0.70z - 205\), making it more straightforward to solve for the variable.

Isolating Variables

Isolating variables is the process of getting the variable alone on one side of the equation to solve for its value. This method is crucial for finding the solution to an equation.
To isolate the variable \(z\) in our example, we first move all terms involving \(z\) to one side.

• Subtract \(0.70z\) from both sides: \(0.65z - 0.70z - 180 = -205\).
• Simplify to find: \(-0.05z - 180 = -205\).

Now that the variable term is isolated, solve for \(z\) by first adding\(180\)to both sides: \(-0.05z = -205 + 180\),which simplifies to \(-0.05z = -25\).
Finally, divide by \(-0.05\)to get \(z = 500\).
Through these steps, the variable is isolated and its value is determined, effectively solving the equation.