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Chapter 2: Problem 24

Solve each equation, and check the solution. $$ 3 m=10 $$

Short Answer

Expert verified
m = \frac{10}{3}

Step by step solution

01

- Isolate the Variable

To solve for the variable m in the equation \(3m = 10\), we need to isolate m. Since m is being multiplied by 3, we can isolate m by dividing both sides of the equation by 3.
02

- Perform Division

Divide both sides of the equation by 3 to get \( m = \frac{10}{3} \). This simplifies to \( m = \frac{10}{3} \).
03

- Check the Solution

To verify the solution, substitute \(m = \frac{10}{3}\) back into the original equation\(3m = 10\). Replace m with \(\frac{10}{3}\): \[ 3 \times \frac{10}{3} = 10 \] Since both sides of the equation equal 10, the solution \(m = \frac{10}{3}\) is correct.

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Key Concepts

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

Isolating the Variable
To start solving the equation, we need to focus on isolating the variable, which means getting the variable 'm' alone on one side of the equation. In the given exercise, the equation is 3m = 10. Here, 'm' is being multiplied by 3. To isolate 'm', we need to do the opposite of multiplication, which is division. By dividing both sides of the equation by 3, we move a step closer to finding the value of 'm'. This step is crucial because it simplifies the equation, making it easier to solve. Remember, the goal is to have 'm' on one side and a numerical value on the other.
Division in Equations
After isolating the variable, the next step involves division. We need to divide both sides of the equation by the same number to maintain equality. In our case, dividing both sides by 3 helps us solve for 'm'.

Here鈥檚 how it works:
- Take the equation 3m = 10.
- Divide both sides by 3: \[\frac{3m}{3} = \frac{10}{3}\].

Simplifying this, we get \[m = \frac{10}{3}\].

This step-by-step division ensures that we correctly solve for 'm'. It's always important to perform the same operation on both sides to keep the equation balanced.
Solution Verification
After arriving at a solution, it鈥檚 important to verify its correctness. This is where substitution comes into play. We take the solution and substitute it back into the original equation to check if both sides are equal.

For our solution 鈥榤 = \( \frac{10}{3} \)鈥�, substitute \( \frac{10}{3} \) back into the original equation 3m = 10:
\[ 3 \times \frac{10}{3} = 10 \].

Here, the multiplication and division cancel each other out, resulting in 10 = 10.
This confirms that our solution is correct.

Verification is a vital step in solving equations. It helps to ensure accuracy and correctness of the solution. Always verify your answers to avoid any mistakes.

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Most popular questions from this chapter

Two of the most watched episodes in television were the final episodes of \(M^{*} A^{*} S^{*} H\) and Cheers. The total number of viewers for these two episodes was about 92 million, with 8 million more people watching the \(M^{*} A^{*} S^{*} H\) episode than the Cheers episode. How many people watched each episode? (Data from Nielsen Media Research.)

Graph the solution set, and write it using interval notation \(6 \leq 3(x-1)<18\)

To see the connection between the solution of an equation and the solutions of the corresponding inequalities. Solve the following equation, and graph the solution set on a number line. \(3 x+2=14\)

If 6 is subtracted from the third of three consecutive odd integers and the result is multiplied by \(2,\) the answer is 23 less than the sum of the first and twice the second of the integers. Find the integers.

Solve each formula for the specified variable. \(A x+B y=C\) for \(x\)
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