Problem 48 - Algebra 1

Question

Solve the equation. $$55-5 y=9 y+27$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 2\).

Step 1: Rearrange the Equation

Bring all terms to one side of the equation: \(55-5y = 9y+27\). This gives us the new equation \(55 - 27 = 9y + 5y\).

Step 2: Simplify the Left and Right Side

Simplify the equation to: \(28 = 14y\).

Step 3: Find the Value of y

Now divide both sides by 14: \(y = 28 / 14 \).

Key Concepts

Linear Equations

Linear equations are equations of the first degree, which means they have no exponent higher than one. They are usually expressed in the form \(ax + b = c\), where \(x\) is the variable to solve for, and \(a\), \(b\), and \(c\) are constants.
Linear equations have a very straightforward structure and their solutions are the point where the equation is satisfied. A real-world analogy is balancing a see-saw by adding or removing weights on either side until it is level.
The example equation, \(55-5y=9y+27\), is a linear equation with \(y\) as the variable we need to solve for.

• Constants and Coefficients: In our exercise, we identify 55 and 27 as constants and -5 and 9 as coefficients of the variable \(y\).
• Goal: The main goal in solving this linear equation is to find the specific value of \(y\) that makes the expression true.

Equation Rearrangement

Rearranging equations is a vital step in solving linear equations. It involves manipulating the equation to bring like terms together. The goal here is to make the equation simple enough to solve.
In our exercise, the equation \(55 - 5y = 9y + 27\) is rearranged by bringing all terms involving \(y\) to one side and constants to the other. This gives us
\(55 - 27 = 9y + 5y\).
Each side of the equation should be simplified to make the solution clearer.

• Step One: Move all terms containing \(y\) to one side. You add \(5y\) to both sides to get all \(y\) terms together.
• Step Two: Simplify the reduced terms on both sides. After moving the terms, subtract \(27\) from \(55\) on the left.

Rearranging simplifies the problem and sets up the next steps in finding the solution.

Algebraic Simplification

Algebraic simplification is about reducing expressions to their simplest form. It's critical when dealing with any equation to make solving easier.
After rearranging, we have the equation \(28 = 14y\). Simplification involves adding, subtracting, multiplying, or dividing terms to isolate the variable.
The simplification process includes:

• Combining Like Terms: In this exercise, \(9y + 5y\) simplifies to \(14y\).
• Isolating the Variable: The next step is dividing both sides by 14 to solve for \(y\). So, \(y = 28 / 14\).

This approach ultimately helps you reach the solution, which, in this case, is \(y = 2\). By understanding these simplification steps, solving even more complex equations can become much easier, fostering your algebraic skills.