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Chapter 10: Problem 44

Solve the equation. Check your solution. $$\frac{1}{2} \mathrm{x}=63$$

Short Answer

Expert verified
The solution to the equation \( \frac{1}{2} \mathrm{x} =63 \) is \(x=126\).

Step by step solution

01

Write out the given equation

The problem provides the equation \( \frac{1}{2} \mathrm{x} =63 \). It is important to understand that the equation represents \(\frac{1}{2} \) of some value x equating to 63.
02

Solve for x

In order to find the value of x that makes the given equation true, we need to isolate x. This can be achieved by multiply both sides of the equation by 2. Doing so, we get \(2*\frac{1}{2} \mathrm{x} =63*2\), which simplifies further to \(x=126\).
03

Check the solution

We can check if the solution is correct by substituting x with 126 in the initial equation. Doing this results in \( \frac{1}{2} * 126 = 63 \), which is indeed a true statement, confirming that x=126 is the correct solution of the given equation.

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Key Concepts

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

Equation Checking
When solving equations, it is crucial to verify that your solution is accurate. This ensures the process carried out was correct and avoids any errors. Once you've found a value for the variable, substitute it back into the original equation. Then, perform the necessary calculations to see if both sides of the equation remain equal.

For example, if you believe that the solution to the equation is \( x = 126 \), plug that into your initial equation: \( \frac{1}{2} \times 126 = 63 \). If the equation holds true, as in this case, you have confirmed your solution.
  • Helps verify calculations
  • Ensures no steps were skipped
  • Builds confidence in problem-solving skills
Multiplication in Equations
Multiplication is often used to eliminate fractions or coefficients when solving equations. It helps simplify equations, making it easier to isolate the variable.

At times, the equations might have a fraction as a coefficient, as in \( \frac{1}{2}x = 63 \), which can be dealt with by multiplying both sides by the reciprocal of the fraction. This effectively clears the fraction. Here, multiplying both sides by 2 gives:
  • \( 2 \times \frac{1}{2}x = 2 \times 63 \)
  • This simplifies to \( x = 126 \)
Multiplication helps simplify and transform equations, making the variable more accessible to solve for.
Isolating Variables
Isolating a variable means getting the variable alone on one side of the equation. This is essential for finding out what value the variable represents.

The process usually involves performing operations that reverse those applied to the variable. If the variable is multiplied by a fraction, for example, you would multiply by the reciprocal to cancel it out.

In the equation \( \frac{1}{2}x = 63 \), isolating \( x \) involves multiplying both sides by 2:
  • This transforms it into \( x = 126 \)
  • Variable is now isolated, providing a clear answer
Remember, each step taken in isolating the variable should be done to both sides of the equation to maintain balance.

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Most popular questions from this chapter

Two chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning.

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