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Chapter 5: Problem 60

Solve and check. $$\frac{2 x}{3}-1=5$$

Short Answer

Expert verified
After solving, the value of x is found to be 9. Upon checking by plugging x=9 back into the original equation, both sides balance confirming that x=9 is the correct solution.

Step by step solution

01

Addition on Both Sides

First, isolate the term that includes x by adding 1 to both sides of the equation. This results in the equation \( \frac{2x}{3} = 5 + 1 \) or \( \frac{2x}{3} = 6 \).
02

Isolation of x

Since the left side of the equation is \( \frac{2x}{3} = 6 \), x can be isolated on one side of the equation by multiplying both sides by the denominator 3. Hence, we get \( 2x = 18 \) after simplification.
03

Solving for x

Now, with \(2x = 18\), x can be completely isolated by dividing both sides by 2, which gives us \(x = 9\)
04

Checking

To confirm the solution, plug x=9 back into the initial equation to see if both sides balance. It becomes \( \frac{2*9}{3}-1 = 5 \), which simplifies to \(6 - 1 = 5\). Both sides of this equation are equal, confirming our solution.

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in mathematics that allows us to find the value of unknown variables that satisfy given mathematical statements. The process involves performing the same operations on both sides of the equation to maintain balance. This helps to simplify the equation until the value of the unknown variable is isolated.
  • Start by identifying the variable and the terms associated with it on each side of the equation.
  • Perform operations such as addition, subtraction, multiplication, or division to both sides equally to simplify the equation.
  • The goal is to perform these operations in a systematic sequence, usually inverse operations, to effectively simplify the equation down to the solution.
By following these steps carefully, you can solve most linear equations with ease.
Isolation of Variable
Isolation of a variable means rearranging the equation so that the unknown variable stands alone on one side of the equation. This is a critical step because it clearly shows the value of the variable.
  • Find the Variable: Initially, locate the term containing the variable on one side of the equation.
  • Remove Constants: Use addition or subtraction to eliminate constants from the side with the variable.
  • Clear Coefficients: Once only a coefficient and the variable are left, divide or multiply to get the variable by itself.
By carefully following these steps, such as adding or subtracting constants first, then focusing on coefficients, you make sure the variable is cleanly isolated. This helps find its precise value.
Checking Solutions
Checking your solutions is just as important as solving the equations. This step ensures that the calculated value truly satisfies the original equation. It acts like a self-check mechanism for your solution.
  • First, substitute the found value of the variable back into the original equation.
  • Calculate both sides separately to check if they are equal.
  • If both sides match, then your solution is verified to be correct.
  • If not, re-examine the steps you took to identify possible mistakes.
This thorough verification confirms the accuracy of your solution and contributes to accurate mathematical practice.

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