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Chapter 2: Problem 48

Solve each equation. Check all solutions. SEE EXAMPLE 4. (OB]ECTIVE 2) $$8 y+4=4 y$$

Short Answer

Expert verified
The solution to the equation is \(y = -1\). This has been checked and confirmed to be valid.

Step by step solution

01

Subtract 4y from both sides

Deduct 4y from both sides of equation to start isolating y. This will result in \(8y - 4y = 4y - 4y + 4\), which simplifies to \(4y + 4 = 0\).
02

Subtract 4 from both sides

To continue isolating y, subtract 4 from both sides which gives \(4y + 4 - 4 = 0 - 4\). This simplifies to \(4y = -4\).
03

Divide by 4

Divide all terms by 4 in order to complete the isolation of y. This results in \(\frac{4y}{4} = \frac{-4}{4}\), which simplifies to \(y = -1\).
04

Check the Solution

This can be done by substituting -1 for y back into the original equation and checking if both sides are equal. Plugging in -1 for y in \(8y + 4 = 4y\) results in \(8*(-1) + 4 ?= 4*(-1)\). This simplifies to \(-8 + 4 ?= -4\), which after simplification gives \(-4 ?= -4\). Both sides are equal, indicating that the solution is valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Variables
When solving linear equations, one of the key strategies is isolating the variable. This means we want to get the variable on one side of the equation all by itself. The goal is to find the value of the variable that makes the equation true. In our example equation, it was important to isolate \( y \) to determine its value.

To start this process:
  • We move all terms containing \( y \) to one side of the equation.
  • This involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
  • The objective is to eliminate all other terms from the side of the variable.
For example, in the equation \( 8y + 4 = 4y \), we subtract \( 4y \) from both sides. This isolates terms with \( y \) on one side while removing \( y \) from the other side.

Remember, keeping the balance of the equation is crucial. Every operation should affect both sides equally to maintain the equation's truth.
Verification of Solutions
After finding a solution, it's always important to verify it. Verification involves substituting the obtained value back into the original equation to ensure that it holds true. This step confirms that your answer is correct and there was no error in your calculations.

In our example, once we solved \( y = -1 \), we substituted this back into the original equation \( 8y + 4 = 4y \).
  • Substitute \( y \) with \( -1 \).
  • Calculate both sides of the equation.
  • If both sides are equal, the solution is verified.
As we saw, substituting \( y = -1 \) results in the equation \( -4 = -4 \), proving our solution is correct. Verification is a simple yet vital step that ensures the integrity of your solution.
Step-by-Step Algebraic Manipulation
To solve equations accurately, it's crucial to follow a methodical, step-by-step approach. Algebraic manipulation involves systematically applying algebraic rules to transform and simplify equations. This method helps avoid mistakes and makes the solution process clear.

For example, in our problem:
  • Step 1: Subtract \( 4y \) from both sides to begin isolating \( y \).
  • Step 2: Subtract 4 from both sides to further isolate \( y \).
  • Step 3: Divide by 4 to solve for \( y \).
  • Step 4: Verify the solution by substituting it back into the original equation.
Each step should be done carefully and checked for accuracy before moving on to the next. Step-by-step algebraic manipulation ensures that we systematically arrive at the right solution, minimizing errors in calculations.

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