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Chapter 10: Problem 43

Solve the equation. Check your solution. $$3 x=145$$

Short Answer

Expert verified
The solution to the equation \(3x = 145\) is \(x \approx 48.33\).

Step by step solution

01

Isolate the Variable

To find the value of \(x\), the equation can be solved by isolating the variable. The equation is \(3x = 145\). To isolate \(x\), divide both sides of the equation by 3.
02

Evaluate \(x\)

That gives \(x = 145 \div 3\).
03

Calculate \(x\)

Evaluating \(145 \div 3\) results in \(x \approx 48.33\).
04

Check Your Solution

Ensure the solution is correct by substituting \(x = 48.33\) into the original equation, i.e., \(3 * 48.33 = 144.99\). The right-hand side is close to 145, confirming that the solution is approximately correct.

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Key Concepts

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

Isolating the Variable
Isolating the variable is a fundamental step in solving linear equations. It entails manipulating the equation to have the variable alone on one side. This process allows us to directly determine the value of the variable. In the equation \(3x = 145\), our goal is to isolate \(x\), which means we want \(x\) to stand alone on one side of the equation.
To achieve this, figure out what operation is combined with \(x\). Here, \(x\) is multiplied by 3. Hence, to isolate \(x\), perform the inverse operation, which is division, on both sides of the equation.
  • Look for the operation attached to \(x\) (multiplication in \(3x\)).
  • Apply the inverse operation (divide both sides by 3).
  • Resulting in \(x = \frac{145}{3}\).
The isolated \(x\) enables us to find its numerical value by simplifying the equation.
Division in Equations
Division is a crucial operation used to solve equations, especially when isolating a variable. In our problem, after determining that \(3x = 145\) needed the right operation, we performed division. This involves dividing both sides of an equation by the same non-zero number, which ensures the equality remains balanced.

For \(3x = 145\):
  • Divide each side by 3 (the coefficient of \(x\)).
  • Express this step as \(x = \frac{145}{3}\), which helps understand the equation's restructuring during division.
  • Calculate \(\frac{145}{3}\) to find the approximate value of \(x\), which is \(x \approx 48.33\).
Remember, dividing both sides by the same number maintains the equality and helps find \(x\)'s actual value in the equation.
Substitution to Check Solution
After solving an equation, always check your answer to ensure accuracy. This involves substituting your obtained solution back into the original equation.

In our example, we solved for \(x\) and got \(x \approx 48.33\). To verify:
  • Substitute \(x\) back into \(3x = 145\).
  • Calculate \(3 \times 48.33\) to see if the left side approximately equals 145.
  • Upon calculation, \(3 \times 48.33 = 144.99\), which is close to 145. This minor difference can be due to rounding.
This process confirms the solution's correctness through validation, ensuring the rounded value satisfies the original equation's conditions.

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