Problem 58 - Algebra A Combined Function

Question

Solve each equation. See Section \(2.3 .\) \(3 z-(4 z-2)=9\)

Step-by-Step Solution

Answer

The solution to the equation is \(z = -7\).

Step 1: Distribute the negative sign

Start by distributing the negative sign in the expression \((4z - 2)\) inside the parentheses. This gives \[3z - 4z + 2 = 9\]

Step 2: Simplify the equation

Combine like terms on the left-hand side of the equation. \(3z - 4z = -1z\) and \(+ 2\) remains, so the equation becomes \[-1z + 2 = 9\]

Step 3: Isolate the variable

Subtract 2 from both sides to isolate the term with the variable \\[-1z = 9 - 2\]Simplifying gives:\[-1z = 7\]

Step 4: Solve for z

Divide both sides by -1 to solve for \(z\):\[z = \frac{7}{-1}\]Which simplifies to:\[z = -7\]

Key Concepts

Linear Equations

Linear equations are fundamental building blocks in algebra. They are equations of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. In our exercise, the equation is presented in a slightly more complex form: \(3z - (4z - 2) = 9\). The goal in solving linear equations is to find the value of the variable that makes the equation true. Here, linear means that each term is either a constant or the product of a constant and a single variable. This makes these equations straightforward to solve through a series of simple algebraic steps.

Linear equations represent straight-line relationships when graphed on a coordinate plane. If you plot these equations, every solution for \(z\) corresponds to a point on the line, making it easy to visualize. Common processes in solving them include simplifying terms, using the properties of equality, and performing operations to both sides of the equation equally.

Problem Solving

Problem solving in algebra involves recognizing and understanding the problem before diving into the solution. It's important to read through the equation carefully and interpret each element.

Begin by identifying key components.

• What are the coefficients and constants in the equation?
• Which operations are being performed?

Solve these types of problems step-by-step:

• First, distribute any constants or variables across parentheses to simplify the expression.
• Next, combine like terms to consolidate the equation into simpler, more manageable parts.
• Finally, embark on isolating the variable to solve the problem fully.

Approaching each step systematically helps prevent errors and solidifies your understanding of the underlying mathematical principles.

Equation Isolation

Equation isolation focuses on getting the variable by itself on one side of the equation. This involves using operations such as addition, subtraction, multiplication, or division to eliminate other terms. The intention is to isolate the variable term on one side with its coefficient.

Here's how to isolate a variable step-by-step:

• Review the equation. Move constant terms to the opposite side by performing the inverse operation. For instance, if you have \(-1z + 2 = 9\), subtract 2 from both sides to get \(-1z = 7\).
• Next, divide or multiply to solve for the variable. In this problem, divide by \(-1\), resulting in \(z = -7\).

This two-step approach ensures that the equation remains balanced, a principle crucial in algebra; whatever you do to one side, you must do to the other.

Mastering equation isolation is critical not only for solving simple linear equations but also for preparing for more advanced algebra topics.