Problem 84 - Algebra 1

Question

Solve the equation. Check the result. (Review 3.4 ) $$8+y=3$$

Step-by-Step Solution

Answer

The solution is \(y = -5\). After checking the result, this solution is found to be correct.

Step 1: Simplifying the equation

Subtract 8 from both sides of the equation. This will leave \(y\) on one side, and \(3 - 8\) on the other, which simplifies to \(3 - 8 = -5\). So, the equation should now read: \(y = -5\).

Step 2: Checking the result

Substitute the value of \(y\) back into the original equation, and check that both sides equal. If they do, then the solving was correct. The original equation is \(8 + y = 3\). By substitifying -5 for \(y\), the equation becomes \(8 + (-5) = 3\), which simplifies to \(3 = 3\). Gives a true statement, so \(y = -5\) is the correct solution.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the value of a variable that makes an equation true. Our goal is to isolate the variable on one side to determine its value. In the equation provided, \(8 + y = 3\), we are tasked with solving for \(y\).

The basic step in solving such equations is to perform operations that will get the variable by itself. You can think of the equation as a balance. Whatever you do to one side, you must do to the other to maintain balance. Here, subtract 8 from both sides:

\[8 + y - 8 = 3 - 8\]

This leaves us with:

\[y = -5\]

When solved correctly, this process reveals the value of \(y\) that satisfies the equation.

Simplifying Equations

Simplifying equations is all about making the equation easier to work with by performing simple arithmetic operations. This often involves combining like terms or eliminating constants from one side to isolate the variable.

In the equation \(8 + y = 3\), our job is to identify what operation will simplify the equation. Subtraction is chosen because we want to zero out the number added to \(y\). By subtracting 8 from both sides, we remove the constant from the left side of the equation:

- Start with \(8 + y = 3\)
- Subtract 8 from both sides: \(y = 3 - 8\)
- Perform the subtraction: \(y = -5\)

This process of simplifying helps in clearly viewing the value of the variable.

Verification of Solutions

Verification of solutions involves checking if the solution you found truly satisfies the original equation. This is an essential step because it confirms that no mistakes were made during calculation.

To verify the solution of \(y = -5\) for the equation \(8 + y = 3\), substitute \(-5\) back into the original equation:

- Replace \(y\) in the equation with \(-5\): \(8 + (-5) = 3\)
- Simplify the left side: \(3 = 3\)

The equation holds true, confirming that our solution \(y = -5\) is correct.

Verification reassures us of accuracy and strengthens understanding of solving equations.