Problem 126 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 5 y+2 y-1=6 y $$

Step-by-Step Solution

Answer

The solution is \( y = 1 \).

Step 1: Combine Like Terms

The given equation is \( 5y + 2y - 1 = 6y \). First, combine the like terms on the left-hand side of the equation. Combine \(5y\) and \(2y\) to obtain \(7y\). The equation simplifies to: \(7y - 1 = 6y\).

Step 2: Isolate the Variable

Next, subtract \(6y\) from both sides of the equation to start isolating the variable \(y\). This will give \(7y - 6y - 1 = 0\). Simplifying further, we get \(y - 1 = 0\).

Step 3: Solve for y

Add 1 to both sides of the equation to completely isolate \(y\). Doing so gives: \(y = 1\). This is the solution for the equation.

Key Concepts

Combining Like Terms

Combining like terms is a fundamental process when working with linear equations. This involves simplifying the equation by merging terms that are similar, which usually means they have the same variable to the same power. In the given exercise, the equation started as \(5y + 2y - 1 = 6y\). Here, both \(5y\) and \(2y\) are like terms because they contain the variable \(y\). By adding these two terms together, we simplify the expression to \(7y - 1 = 6y\).
This process helps reduce the complexity of the equation, making it easier to isolate the variable in the following steps. Always look for coefficients with the same variable as an initial step to streamline your work.

Isolating Variables

Isolating the variable is the next crucial step in solving linear equations. This involves rearranging the equation so that the variable you are solving for stands alone on one side of the equation, usually the left side. In our exercise, we began with the equation \(7y - 1 = 6y\).
In order to isolate \(y\), we first subtract \(6y\) from both sides. After performing this operation, the equation becomes \(7y - 6y - 1 = 0\), which simplifies to \(y - 1 = 0\). This manipulation leaves us with a simpler equation where the variable \(y\) is more prominently featured and almost isolated for the final step.
Always keep in mind that reversing operations, like addition and subtraction, is often necessary to achieve this simplified form.

Solving Equations

Solving the equation is the final step where you find the value of the variable. Once like terms have been combined and the variable isolated, you're usually left with a straightforward equation to solve. In this exercise, after isolating \(y\), we were left with the equation \(y - 1 = 0\).
To solve for the variable \(y\), we simply perform the inverse operation of subtracting 1, which is to add 1 to both sides of the equation. This balances the equation and results in \(y = 1\).
By following these methodical steps, you ensure that all operations are balanced, leading you to a correct solution. Always review your work to confirm that the solution satisfies the original equation.