Problem 7 - Algebra 2

Question

Solve each equation. Check your solution. $$ y+14=-7 $$

Step-by-Step Solution

Answer

The solution is \(y = -21\).

Step 1: Isolate the variable

Start by moving 14 to the other side of the equation. To do this, you subtract 14 from both sides:\[y + 14 - 14 = -7 - 14\]This simplifies to:\[y = -21\]

Step 2: Verify the solution

Substitute \(y = -21\) back into the original equation to check if it's correct:\[-21 + 14 = -7\]On simplification, \(-21 + 14 = -7\), so the solution is verified as correct.

Key Concepts

Isolation of Variable

When solving linear equations, one of the primary tasks is to isolate the variable you are trying to solve for. The goal here is to have the variable on one side of the equation and the constant values on the other.
This means making it the subject of the equation. In the exercise example, "y + 14 = -7", our task is to isolate the variable "y".
To do this, follow these steps:

• First, identify any constants that are added or subtracted from the variable. Here it's the "+14" adjacent to "y".
• Next, perform the opposite operation to both sides of the equation to eliminate these constants.
• In our example, "-14" is subtracted from both sides to balance the equation and isolate "y": \[ y + 14 - 14 = -7 - 14 \]

This operation leaves you with the variable on one side, resulting in: \[ y = -21 \]
By isolating "y", we have found our solution for the variable.

Checking Solutions

Once you have isolated the variable and found a potential solution, it's crucial to check that this solution is correct. This ensures that the solution satisfies the original equation.
Checking solutions involves substituting the found value back into the original equation.
For our example, we found "y = -21". Follow these steps to verify if this solution is correct:

• Take the solution \( y = -21 \) and substitute it back into the original equation \( y + 14 = -7 \).
• By replacing "y" with "-21":
  \( -21 + 14 = -7 \)
• Simplifying gives \( -7 = -7 \), showing that both sides of the equation balance.

This balance confirms that our solution is indeed correct. Checking your solutions helps in catching any arithmetic errors made during the isolation process.

Substitution Method

The substitution method is a key technique in solving equations, particularly useful when verifying solutions or tackling more complex systems of equations. Although in our specific example it's not employed to find the solution, it becomes essential to verify or cross-check the result.
Here's a quick breakdown of how to apply it:

• When you solve for a variable and get a result like "y = -21", this value can be substituted back into another equation if you're dealing with a system.
• Even in a single equation solution, you substitute "y = -21" back into the main equation to ensure all operations hold true, as shown in the checking step.
• Substitution is fundamentally about replacing a variable with its solved value and checking the consistency of the equation or system.

The method is straightforward but powerful, especially when combined with isolation and verification. It's a tool that confirms not just the solution's correctness, but also develops an understanding of how solutions relate to entered equations.