Problem 42 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(-17 z-4=-16 z-20\)

Step-by-Step Solution

Answer

The solution is \(z = 16\).

Step 1: Simplify both sides

Look at each side of the equation: \(-17z - 4 = -16z - 20\). Both sides are already simplified, so we don't need to do anything in this step.

Step 2: Isolate the variable term

We need to get all terms involving \(z\) on one side of the equation. To do this, add \(17z\) to both sides: \(-17z - 4 + 17z = -16z - 20 + 17z\).This simplifies to: \(-4 = z - 20\).

Step 3: Solve for the variable

Now, we isolate \(z\) by adding \(20\) to both sides of the equation:\(-4 + 20 = z - 20 + 20\).This simplifies to: \(16 = z\).

Step 4: Verification

Substitute \(z = 16\) back into the original equation to verify the solution:\(-17(16) - 4 = -16(16) - 20\).Calculate both sides: \(-272 - 4 = -276\) and \(-256 - 20 = -276\). Both sides are equal, so \(z = 16\) is the correct solution.

Key Concepts

Equation Simplification

Equation simplification involves making an equation easier to solve by combining like terms or removing unnecessary parts. In algebra, this might mean collecting terms on one side or reducing expressions when possible.

Even though in the original exercise the equation was already simplified, understanding what simplification entails is crucial:

• Identify and combine any like terms on both sides of the equation, such as similar variables or constant numbers.
• Remove any extraneous numbers by adding or subtracting them to/from both sides to make the equation less cluttered.

Always start by asking yourself whether any terms can be merged or reduced, which often simplifies finding the solution later. Typically, simplifying the equation makes the subsequent steps, like isolating the variable, much more manageable.

Isolation of Variables

To solve algebraic equations, isolating the variable is a crucial step. This is about transforming the equation to get the variable alone on one side.

Here's how you can isolate the variable step by step:

• First, move all terms containing the variable to one side of the equation. You can do this by adding or subtracting terms. This often involves reversing addition or subtraction operations. For instance, if you have \( -17z = -16z + 20\), add \(17z\) to both sides to cancel it out on one side.
• Next, handle any coefficients. If the variable has coefficients (numbers multiplied by the variable), divide or multiply to get the variable by itself.
• Finally, adjust any constants. If there are numbers added or subtracted from the variable, perform the opposite operation to both sides.

Once you have the variable isolated, you can easily write the solution for the variable. This method is systematic and helps you break down complex equations into simpler parts.

Verification of Solutions

Verification is your mathematical double-check to ensure that the solution actually solves the equation. Start by substituting the solution back into the original equation. This step confirms if both sides of the equation balance, confirming that no mistakes were made.

For the exercise given:

• Plug the solution \( z = 16 \) back into the equation \( -17z - 4 = -16z - 20 \).
• Calculate each side separately. For example, the left side becomes \( -17 imes 16 - 4\) leading to \(-276\).
• The right side becomes \( -16 imes 16 - 20\) also simplifying to \(-276\).
• Because both sides equal, the solution is verified.

Verification acts as a safeguard against errors, ensuring the accuracy of your solution.