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Chapter 9: Problem 74

In Exercises \(71-74,\) solve the given equation. $$0.25 m+0.1 m=56$$

Short Answer

Expert verified
The solution is \( m = 160 \).

Step by step solution

01

- Simplify the Equation

Combine the like terms on the left side of the equation: \[0.25m + 0.1m = 0.35m\]This simplifies the equation to:\[0.35m = 56\]
02

- Isolate the Variable

Divide both sides of the equation by 0.35 to isolate the variable \( m \):\[m = \frac{56}{0.35}\]
03

- Simplify the Expression

Perform the division to find the value of \( m \):\[m = 160\]

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Key Concepts

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

Combining Like Terms
In the realm of linear equations, combining like terms is a crucial step to simplify the equation. Like terms are terms that contain the same variable raised to the same power. For example, in the equation as given in the exercise, both terms on the left side, 0.25m and 0.1m, are considered like terms because they both contain the variable 'm'.

Combining them simplifies our equation and makes it easier to solve. Here's how it works:
  • Identify the like terms, which in this case are 0.25m and 0.1m.
  • Add their coefficients: 0.25 + 0.1 gives us 0.35.
  • Rewrite the equation with the combined terms: 0.35m = 56.
Combining like terms reduces the complexity of the equation, which is the first step in finding the solution.
Isolating the Variable
Once we have combined like terms, the next step is to isolate the variable. Isolating the variable means getting it alone on one side of the equation, which helps us to easily solve for its value.

In our simplified equation 0.35m = 56, we need to get 'm' by itself. Here's how we do it:
  • We divide both sides of the equation by the coefficient of 'm', which is 0.35.
  • This operation ensures that 'm' is isolated, as dividing 0.35m by 0.35 leaves us with just 'm'.
  • Hence, the equation becomes m = 56 / 0.35.
By isolating the variable, we have reached a point where m is alone, making it easy to further calculate its value.
Simplifying Expressions
The final piece of the puzzle is to simplify the expression to find the value of the variable. Simplifying here involves performing the arithmetic operation of division.

With our equation m = 56 / 0.35:
  • We perform the division: 56 divided by 0.35.
  • This calculation yields m = 160.
Hence, the value of 'm' is simplified and found to be 160.

Each of these steps - combining like terms, isolating the variable, and simplifying expressions - builds upon the previous one to solve the equation efficiently and correctly.

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Most popular questions from this chapter

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$A=\frac{1}{2} h\left(b_{1}+b_{2}\right) \quad \text { for } h$$

Solve each of the problems algebraically. Be sure to label what the variable represents. In a certain voting district the ratio of Republicans to Democrats is 4 to \(7 .\) If there are 2520 Democrats, how many Republicans are there?

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. \(\frac{x-\mu}{s}<2\) for \(x \quad\) (Assume that \(s>0\) )

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$a(b+c)=d \quad \text { for } a$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{5}{x^{2}-2 x-3}=\frac{4}{x^{2}-3 x-4}$$
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