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Chapter 2: Problem 7

Solve each equation using the Subtraction and Addition Properties of Equality. $$y+45=-66$$

Short Answer

Expert verified
y = -111

Step by step solution

01

- Understand the equation

The given equation is: \[ y + 45 = -66 \]
02

- Isolate the variable

To isolate the variable \( y \), apply the Subtraction Property of Equality by subtracting 45 from both sides of the equation:\[ y + 45 - 45 = -66 - 45 \]
03

- Simplify both sides

Simplify both sides of the equation:\[ y = -111 \]The solution is now found.

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Key Concepts

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

Subtraction Property of Equality
To solve linear equations effectively, it鈥檚 crucial to understand the Subtraction Property of Equality. This property allows you to maintain balance in an equation by subtracting the same amount from both sides. This consistency ensures the equality holds.
When you have an equation like $$y + 45 = -66$$, you aim to isolate the variable. Here, the addition of 45 to y is complicating things. By subtracting 45 from both sides鈥攃ourtesy of the Subtraction Property鈥攜ou can begin resolving this. It leads to:
\[ y + 45 - 45 = -66 - 45 \]\ The elimination of 45 on the left simplifies things, giving you a clearer path to solve for y.
Isolation of Variables
Isolating the variable is one of the key steps in solving linear equations. It means getting the variable on one side of the equation. To do this, we use inverse operations.
In the equation $$y + 45 = -66$$, your goal is to have y alone on one side. Subtraction of 45 from both sides is a key step. This isolation moves the 'noise' or number on the same side as y to the other side. Here's how it looks: \[ y + 45 - 45 = -66 - 45 \]\
After simplifying, y is isolated as \[ y = -111 \]\. Now, you have successfully isolated the variable. Simple, right?
Simplifying Equations
Once you've applied the Subtraction Property and isolated the variable, the final step is simplifying the equation. This means performing the arithmetic operations needed to solve for the variable.
From the earlier step, you have: $$ y = -66 - 45 $$. To solve completely, simplify the right side: \[ y = -111 \]\.
Once simplified, it's easy to see the solution to the original equation. Simplifying equations is like cleaning up clutter鈥攜ou focus on the essential elements to reach the answer.
  • Ensure all like terms are combined.
  • Perform arithmetic carefully to avoid errors.
  • Double-check the simplification to confirm accuracy.

Understanding each of these steps leads to mastering linear equations effortlessly!

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