Problem 12 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each solution. $$ 5 y+8 y-11=-11 $$

Step-by-Step Solution

Answer

The solution is \(y = 0\). It satisfies the original equation.

Step 1: Combine Like Terms

Start by combining the terms on the left-hand side of the equation that contain the variable. The equation is \(5y + 8y - 11 = -11\). Combine the \(5y\) and \(8y\) to simplify the expression to \(13y - 11 = -11\).

Step 2: Isolate the Variable Term

The next step is to isolate the term that contains \(y\). Add \(11\) to both sides of the equation to eliminate the constant on the left, resulting in \(13y = 0\).

Step 3: Solve for the Variable

Now that you have \(13y = 0\), divide both sides by \(13\) to solve for \(y\). This gives you \(y = 0\).

Step 4: Check the Solution

Substitute \(y = 0\) back into the original equation to verify if it's correct: \(5(0) + 8(0) - 11 = -11\). Simplifying gives \(-11 = -11\), which is true, so the solution is verified.

Key Concepts

Combining Like Terms

When solving linear equations, one fundamental step is combining like terms. This means simplifying expressions by adding or subtracting terms that have the same variable or are constants. In the equation provided,

• We first look at the left-hand side: \(5y + 8y - 11\).
• The terms \(5y\) and \(8y\) are like terms because they both contain the variable \(y\). We can add these coefficients together.
• This results in \(13y\).

By combining \(5y\) and \(8y\), the expression simplifies, making it easier to further solve the equation. The key here is focusing on terms with the same variables and then performing the arithmetic operation, in this case, addition.

Isolating Variables

Isolating the variable is a critical step to finding its value in an equation. Once you have combined like terms, the next goal is to make the variable stand alone on one side of the equation.

For the equation \(13y - 11 = -11\):

• You begin by removing constants from the left side to have \(y\) by itself. Here, add \(11\) to both sides to cancel out the \(-11\).
• The equation becomes \(13y = 0\).

The aim is to ensure only the variable, in this case \(y\), remains, so it's solely on one side. Once that's done, solve by dividing both sides by the coefficient of \(y\), which is \(13\).
This gives \(y = 0\). This step requires careful arithmetic to ensure the variable is truly isolated without changing the equation's balance.

Checking Solutions

Once you solve for the variable, it's vital to check your solution. This confirms that your solution is correct and that you didn't make any errors in your calculations.

For our original equation: \(5y + 8y - 11 = -11\) and the solution \(y = 0\):

• Substitute \(0\) for \(y\) back into the equation.
• Calculate: \(5(0) + 8(0) - 11\).
• This simplifies to \(-11 = -11\), a true statement.

When both sides of the equation are equal after substitution, it verifies your solution. This is an essential habit since it ensures the solution holds true to the original equation. Always double-check your work with this method to ensure accuracy and build confidence in solving equations.