Problem 104 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 6 y-11=13 $$

Step-by-Step Solution

Answer

The solution is \( y = 4 \).

Step 1: Add 11 to both sides

Start by isolating the variable term on one side. To do this, add 11 to both sides of the equation. This will eliminate the -11 from the left side:\[6y - 11 + 11 = 13 + 11\] Simplify both sides:\[6y = 24\]

Step 2: Divide both sides by 6

Now that the equation is \(6y = 24\), isolate \(y\) by dividing both sides by 6:\[\frac{6y}{6} = \frac{24}{6}\]Simplify the division:\[y = 4\]

Key Concepts

Isolation of Variables

When solving linear equations, the goal is to find the value of the variable that makes the equation true. This process often requires isolating the variable, which means getting the variable on one side of the equation and everything else on the other. It usually involves rearranging and manipulating the equation using arithmetic operations.

In the example provided, we have the equation \(6y - 11 = 13\). Our first step to isolating the variable \(y\) is to eliminate any constants on the side of the equation where the variable is present. We do this to "free" the variable from any numbers it's being added to or subtracted by. Here, adding 11 to both sides cancels out the -11, leaving \(6y\) by itself. The equation then simplifies to \(6y = 24\). By focusing on isolating the variable, we simplify the equation and bring it closer to solving for \(y\).

Arithmetic Operations

Arithmetic operations are fundamental to solving equations. In essence, they include the well-known operations: addition, subtraction, multiplication, and division. By using these operations strategically, we can manipulate equations to find the value of the variable.

In the step-by-step solution you saw earlier, we first used addition. By adding 11 to both sides, we eliminated a constant from where \(y\) is located in the equation. After that, we used division, which is another essential arithmetic operation, to further simplify the equation. We divided both sides by 6 to solve for \(y\). These operations respect the principle of balance in equations, meaning whatever operation you do to one side, you must do to the other. This keeps the equation equivalent throughout the solving process.

Simplification in Equations

Simplification is a vital step in solving equations which focuses on making the equation as straightforward as possible. This might involve combining like terms, canceling out terms, or breaking down complex expressions into simpler parts.

In our example problem, after performing the necessary arithmetic operations like addition and division, we reached a stage where further simplification made it possible to identify the value of the variable. By simplifying terms on both sides, we derived the simplest form of the equation, which is \(y = 4\). Breaking down equations through simplification makes the value of the variable clear and confirmable.

• Combine like terms whenever possible to reduce the number of elements in the equation.
• The aim of simplification is to achieve an easy-to-read and -understand form of the equation that directly leads to finding the solution.