Problem 52 - Algebra 2

Question

Solve each equation. Check your solution. $$ 2 z-3=-6 z+1 $$

Step-by-Step Solution

Answer

The solution is \(z = \frac{1}{2}\).

Step 1: Add 6z to Both Sides

Start by eliminating the variable on one side of the equation. Add \(6z\) to both sides to get all the variables on one side:\[2z + 6z - 3 = 1\]

Step 2: Simplify Both Sides

Combine like terms on both sides:\[8z - 3 = 1\]

Step 3: Add 3 to Both Sides

To isolate the term with the variable, add 3 to both sides:\[8z - 3 + 3 = 1 + 3\]This simplifies to:\[8z = 4\]

Step 4: Divide by 8

To solve for \(z\), divide both sides by 8:\[z = \frac{4}{8}\]Simplifying the fraction gives:\[z = \frac{1}{2}\]

Step 5: Check Your Solution

Substitute \(z = \frac{1}{2}\) back into the original equation:\[2 \left(\frac{1}{2}\right) - 3 = -6 \left(\frac{1}{2}\right) + 1\]Calculate both sides:\[1 - 3 = -3 + 1\]\[-2 = -2\]Since both sides are equal, the solution \(z = \frac{1}{2}\) is correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is fundamental when solving linear equations. It allows us to rearrange and simplify equations to isolate variables effectively.
Let's take a closer look at how we manipulate equations:

• **Balancing an Equation:** Always perform the same operation on both sides to keep the equation balanced. This could be adding, subtracting, multiplying, or dividing.
• **Example:** In our exercise, moving terms like \(6z\) ensures all variable terms are consolidated on one side. \(2z + 6z - 3 = 1\).
• **Purpose:** The manipulation drives us toward a simpler equation, making our variable more accessible for isolation.

By consistently applying these operations, the complex equation gradually becomes solvable.

Checking Solutions

Checking solutions is an essential step in verifying the accuracy of your result when solving an equation. This step ensures that the obtained solution satisfies the original equation.
Here's how you can check your solution effectively:

• **Substitution:** Substitute the solution back into the original equation. For our problem, we substitute \(z = \frac{1}{2}\).
• **Simplify Both Sides:** Carry out the operations to ensure both sides of the equation are equal.
  For this exercise, plugging our solution back, \(2 \left( \frac{1}{2} \right) - 3 = -6 \left( \frac{1}{2} \right) + 1\), becomes \(-2 = -2\), confirming equality.
• **Confirm Equality:** When both sides are equal, your solution is correct!

This step not only boosts confidence but firmly verifies your entire solution process.

Variable Isolation

Variable isolation involves getting the variable alone on one side of the equation. This is a key step in solving equations.
Let's explore the process:

• **Goal:** Move all terms without the variable to the other side. For example, after simplifying, we have \(8z = 4\).
• **Operations Used:** Use operations like addition and subtraction to eliminate constants, and division to remove coefficients.
• **Final Step:** Divide by the coefficient of the variable. In our case, divide both sides by \(8\) resulting in \(z = \frac{1}{2}\).

Isolating the variable transforms the equation into a straightforward form, ready to tell us the value of the variable.

Equation Simplification

Simplifying the equation means reducing it to its simplest form without losing any information. This process involves several important steps.
Here’s how simplification occurs:

• **Combine Like Terms:** Initially, combine terms such as \(2z + 6z\) to \(8z\).
• **Remove Constants:** Use operations like addition to eliminate constants from one side, leading to simpler expressions like \(8z = 4\).
• **Fraction Simplification:** After solving, often further simplify fractions if possible – \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\).

By condensing terms and removing unnecessary complexity, the equation becomes clear and concise, making it much easier to solve and understand.