Problem 68 - Elementary Algebra Within Reach

Question

What value of \(y\) would cause \(3 y+2\) to equal 8 ? Explain.

Step-by-Step Solution

Answer

The value of \(y\) that will make the expression \(3y+2\) equal to 8 is \(y = 2\).

Step 1: Understand the problem

Before starting to solve the equation, it is crucial to understand it. Here, we are given the equation \(3y+2 = 8\). We can notice that it's a linear equation, and we need to find the value of variable \(y\) that will make this equation true.

Step 2: Isolating the variable

To isolate variable \(y\), we'll start by canceling out the '+2' on the left side. We can do this by subtracting 2 from both sides of the equation \(3y + 2 - 2 = 8 - 2\), simplifying to \(3y = 6\).

Step 3: Solve for y

Finally, to solve for \(y\) we can divide both sides of the equation by 3. That leads us to \(y = \frac{6}{3}\), which simplifies to \(y = 2\).

Key Concepts

Solving Equations

Linear equations are a fundamental part of algebra, representing relationships using constants and variables. Here, solving equations involves finding values for the unknown variables that make the equation true.
For the given problem, we have the equation: \[ 3y + 2 = 8 \]To solve it, we aim to find the value of the variable, which in this case is \(y\). Solving this equation requires performing operations that simplify it progressively while keeping the equation balanced.

• First, understand the equation structure and goal: to isolate \(y\).
• Use inverse operations, such as subtraction and division, to simplify equations.

Grasping how to solve linear equations helps in understanding more complex equations and real-world problems.

Isolating Variables

Isolating the variable is an essential step in solving equations. It involves manipulating the equation to get the variable by itself on one side. This helps in identifying its value directly.
In our equation, \[ 3y + 2 = 8 \]we first focus on removing the constant term from the \(y\)'s side. We do this by subtracting \(2\) from both sides, yielding: \[ 3y = 6 \]This step simplifies the equation significantly, leaving a clearer path to the solution.
Remember:

• Whatever action you perform on one side of the equation must also be applied to the other side.
• Aim to have the variable's coefficient as 1 to make it easy to identify the variable's value.

Isolating the variable aligns with the principle of maintaining equality.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that involves understanding problems, devising a strategy, and applying methods to find solutions. The linear equation example provides a simplistic view of these steps.
When tackling math problems:

• First, interpret the problem completely.
• Choose strategies like isolating variables that simplify the process.
• Apply logical operations and check the solution's validity by plugging it back into the original equation.

In the case solution, after we have simplified to \[ 3y = 6 \]we solve for \(y\) by performing division, giving us \[ y = \frac{6}{3} = 2 \]By focusing on these steps, you enhance your mathematical reasoning and analytical thinking.