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Chapter 3: Problem 14

solve the given equation. If the equation is always true or has no solution, indicate this. $$3 a+7=10 a-1$$

Short Answer

Expert verified
The solution is \(x = 6\).

Step by step solution

01

Identify the Goal

Determine the value of the variable, which in this case is to solve for \(x\) in the equation \(8x = 48\).
02

Isolate the Variable

Divide both sides of the equation by 8 to isolate \(x\): \[ x = \frac{48}{8} \]
03

Simplify the Equation

Perform the division to find the value of \(x\): \[ x = 6 \]
04

Verify the Solution

Substitute \(x = 6\) back into the original equation to ensure it holds true: \[ 8 \times 6 = 48 \] This confirms that \(8x = 48\) is satisfied when \(x = 6\).

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Key Concepts

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

Isolating Variables
In the process of solving linear equations, the first important step is isolating the variable. The goal here is to get the variable on one side of the equation and everything else on the other side.

To isolate the variable in the given equation, \(8x = 48\), we need to perform an operation that leaves only \(x\) on one side.

Here's how you can do it:
  • Identify the coefficient of the variable (which is 8 in this case).
  • Divide both sides of the equation by this coefficient.
  • This gives us \[x = \frac{48}{8}\].
By dividing both sides by 8, \(x\) is isolated, making it the subject of the formula.
Simplifying Equations
Once the variable is isolated, the next step is to simplify the equation to find its value. Simplifying involves performing the actual arithmetic operations to determine the value of the isolated variable.

In the previous step, we isolated \(x\) by dividing both sides by 8. This gives us:
  • \[x = \frac{48}{8}\]
Now perform the division to simplify and find the value of \(x\):
  • Divide 48 by 8
  • Resulting in \[x = 6\]
So, the simplified equation tells us that \(x = 6\).
Verifying Solutions
After finding the value of the variable, it鈥檚 crucial to verify the solution. Verification ensures that the solution is correct and satisfies the original equation.

To verify the solution for the given equation \(8x = 48\), follow these steps:
  • Substitute the found value of \(x\) (which is 6) back into the original equation.
  • Check if the left-hand side equals the right-hand side:
  • \[8 \times 6 = 48\]
Since \(48 = 48\) is a true statement, the solution \(x = 6\) is verified.
This confirms that our solution is correct and that the original equation holds true when \(x = 6\). Remember to always verify your solutions to ensure their accuracy.

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Most popular questions from this chapter

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