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Chapter 2: Problem 8

Solve each equation. \(6 y-7=41\)

Short Answer

Expert verified
The solution is \(y = 8\).

Step by step solution

01

Understand the Goal

The objective is to isolate the variable \(y\) on one side of the equation to find its value.
02

Add 7 to Both Sides

To start isolating \(y\), eliminate the \(-7\) on the left side. Add 7 to both sides of the equation: \[6y - 7 + 7 = 41 + 7\]This simplifies to:\[6y = 48\]
03

Divide by 6

Now that we have \(6y = 48\), divide both sides by 6 to solve for \(y\): \[\frac{6y}{6} = \frac{48}{6}\]This simplifies to:\[y = 8\]
04

Verify Solution

To ensure the solution is correct, substitute \(y = 8\) back into the original equation and check: \[6(8) - 7 = 48 - 7 = 41\]Since both sides of the equation are equal, \(y = 8\) is verified as correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
In math, isolating variables is like solving a mystery. Our goal is to figure out what the variable stands for by itself. Let鈥檚 use our example from the exercise, \(6y - 7 = 41\), to understand this process.
  • Start by identifying the variable in the equation, which is \(y\) in our case.
  • The variable \(y\) is currently trapped with other numbers. We need to get \(y\) alone on one side of the equation.
  • This involves performing inverse operations to eliminate the other numbers. Here, we do not want the \(-7\) on the left side with \(y\), so we add 7 to both sides to cancel it out.
By doing this, we make the equation simpler, getting closer to knowing what \(y\) exactly is. After simplifying, our equation becomes \(6y = 48\). The next step is to further simplify and isolate \(y\) completely.
Equation Solving Steps
Solving an equation is a step-by-step journey. Once we have simplified the equation to \(6y = 48\), our task now is to solve for \(y\). Let's follow the clues through the solving steps:
  • The equation \(6y = 48\) tells us that 6 groups of \(y\) equals 48. To discover what a single group (or just \(y\)) is, divide both sides of the equation by 6.
  • This gives us \(y = \frac{48}{6}\).
  • Performing the division on the right side reveals \(y = 8\).
Think of this process like unwrapping a gift. Each operation unwraps more of the layers hiding the variable. By performing these steps, you successfully solve for the variable and complete the equation.
Verification of Solutions
Verifying the solution is like double-checking your work in math. It ensures that the value you found is correct. For our equation, we found \(y = 8\). Here's how to verify:
  • Take the solution \(y = 8\) and substitute it back into the original equation: \(6 \times 8 - 7\).
  • Calculate the left side of the equation: \(48 - 7 = 41\).
  • If this equals the right side (41), your solution is verified.
Verification is crucial because it confirms that no mistakes were made during calculations. This method can be applied to any equation, ensuring accuracy and confidence in your mathematical solutions.

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Most popular questions from this chapter

Solve each equation and inequality. \(|x-2|>6\)

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