Problem 31 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 32=4 y+6 $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{13}{2} \).

Step 1: Isolate the variable term

Subtract 6 from both sides of the equation to isolate the variable term.\[ 32 - 6 = 4y \] Which simplifies to: \[ 26 = 4y \]

Step 2: Solve for the variable

Divide both sides of the equation by 4 to solve for \( y \).\[ \frac{26}{4} = y \] Simplifying the fraction gives: \[ y = \frac{13}{2} \]

Step 3: Check the result

Substitute the value of \( y \) back into the original equation to verify the solution.\[ 32 = 4\left(\frac{13}{2}\right) + 6 \] Calculate the right side: \[ 4 \times \frac{13}{2} = 26 \] Thus, \[ 26 + 6 = 32 \] which matches the left side of the original equation, confirming that \( y = \frac{13}{2} \) is correct.

Key Concepts

Isolation of Variables

In solving linear equations, one of the primary steps is the isolation of the variable. This means you want to get the variable by itself on one side of the equation. To do this effectively:

• Identify the variable term you want to isolate (in this case, the term with \( y \)).
• Perform operations that will help "move" everything else to the other side of the equation.
• The goal is to have the variable on one side and all constants on the other.

For example, in the equation \( 32 = 4y + 6 \), you subtract 6 from both sides to start isolating \( y \). This leaves you with \( 26 = 4y \). By completing this step, you've set up the equation so that solving for \( y \) becomes more straightforward. This method works for all basic linear equations, regardless of the variable or constants involved. The idea is to "undo" any added or multiplied numbers attached to the variable by applying opposite operations.

Checking Solutions

Once a potential solution to an equation is found, it's crucial to check that it actually satisfies the equation. This involves substituting the found variable value back into the original equation and verifying that both sides equal each other.Here's how you can do it:

• Take your calculated value of the variable (in this case, \( y = \frac{13}{2} \)).
• Substitute it back into the original equation: \( 32 = 4y + 6 \).
• Calculate to see if both sides of the equation remain equal.
  • If they are equal, then the solution is correct.
  • If not, recheck your steps for any possible errors.

For our example, substituting \( y = \frac{13}{2} \) gives \( 4 \times \frac{13}{2} + 6 = 32 \). Calculate the right-hand side to confirm that it equals 32, validating that your solution is accurate. Checking ensures accuracy and builds confidence in the answer.

Simplifying Equations

Simplifying equations means making them as straightforward as possible, which often involves breaking down complex expressions into simpler components. This process can involve reducing fractions, combining like terms, or removing parentheses by distributing.For instance:

• When you divide both sides of an equation like \( 26 = 4y \) by 4, you aim to simplify \( y \) on one side: \( y = \frac{26}{4} \).
• Further simplify the fraction if possible. In this example, \( \frac{26}{4} \) simplifies to \( \frac{13}{2} \), which is a cleaner and more compact expression for \( y \).

Simplifying the equation at each step helps in minimizing errors and makes it easier to check results later. It means harnessing the most straightforward form of numbers and expressions, aiding both in calculation and comprehension. With practice, simplifying becomes an intuitive part of solving equations.