Problem 8 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$3 y+5=9 y+8$$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{1}{2} \).

Step 1: Identify the Equation

The given equation is \(3y + 5 = 9y + 8\). Our goal is to solve for \(y\).

Step 2: Rearrange to Isolate Variable Terms

Subtract \(3y\) from both sides to have all terms involving \(y\) on one side. \\[ 5 = 6y + 8 \]

Step 3: Isolate the Constant

Subtract 8 from both sides to isolate the term with the variable. \\[ 5 - 8 = 6y \] \\[ -3 = 6y \]

Step 4: Solve for \(y\)

Divide both sides by 6 to solve for \(y\). \\[ y = -\frac{1}{2} \]

Step 5: Verify the Solution

Substitute \(y = -\frac{1}{2}\) back into the original equation to ensure both sides are equal. \Original: \( 3(-\frac{1}{2}) + 5 = 9(-\frac{1}{2}) + 8 \) \Simplified Left: \( -\frac{3}{2} + 5 = \frac{7}{2} \) \Simplified Right: \( -\frac{9}{2} + 8 = \frac{7}{2} \) \Since both sides equal, the solution is verified.

Key Concepts

Isolating Variables

Isolating variables is a fundamental skill in solving linear equations. It means simplifying the equation so that the variable you want to solve for is on one side of the equation by itself. Consider this example equation, \(3y + 5 = 9y + 8\). Here, our goal is to have \(y\) alone on one side.
To begin, identify and move all terms with \(y\) to one side. This involves subtracting \(3y\) from both sides, giving us \(5 = 6y + 8\). Now, see that all \(y\) related terms are on the right.
Next, we need to get rid of the constant next to \(y\), by moving it to the other side. This is done by subtracting \(8\) from both sides, resulting in \(-3 = 6y\). This step involves moving numbers around such that the variable stands alone. Once achieved, it's much easier to find the value of \(y\) by further simplifying.

Verifying Solutions

Verification is an essential step in solving equations as it confirms that the solution is indeed correct. After finding a possible solution for the variable, substitute it back into the original equation.
In our example, after solving, we arrived at \(y = -\frac{1}{2}\). To verify, substitute \(-\frac{1}{2}\) in place of \(y\) in both sides of the original equation. You get:

• Left Side: \(3(-\frac{1}{2}) + 5 = \frac{7}{2}\)
• Right Side: \(9(-\frac{1}{2}) + 8 = \frac{7}{2}\)

Since both sides equal \(\frac{7}{2}\), the solution \(y = -\frac{1}{2}\) is correct, proving the answer satisfies the initial equation.
This step ensures no mistakes were made during manipulation of terms and gives you confidence in your solution.

Step-by-Step Solutions

Adopting a step-by-step strategy for solving linear equations helps prevent errors and clarifies your thought process.
The methodical approach involves:

• **Identifying the equation**: Start by writing down the equation clearly.
• **Rearranging**: Move terms around to have the variable terms on one side. This could involve using addition, subtraction, or both.
• **Simplifying**: Solve for the variable by performing arithmetic operations like division.
• **Verification**: Always plug the solution back into the original equation to verify.

This approach not only aids in solving the current problem but enhances your problem-solving skills for future equations. Clear steps create a roadmap that leads to the right solution effortlessly.