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Chapter 4: Problem 40

Solve the equations and inequalities. $$0.65 m+1.5 m=0.43$$

Short Answer

Expert verified
m = 0.2

Step by step solution

01

Combine like terms

Combine the terms involving the variable \(m\). The equation \(0.65m + 1.5m = 0.43\) simplifies to \((0.65 + 1.5)m = 0.43\).
02

Simplify the coefficients

Add the coefficients of \(m\) together: \(0.65 + 1.5 = 2.15\). The equation now is \(2.15m = 0.43\).
03

Isolate the variable

Divide both sides of the equation by the coefficient of \(m\), which is \(2.15\): \[ m = \frac{0.43}{2.15} \]
04

Calculate the solution

Perform the division to find the value of \(m\): \[ m = 0.2 \]

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Key Concepts

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

Combining Like Terms
In algebra, combining like terms is an essential skill. It involves adding or subtracting terms that have the same variable and exponent.
For example, in our original equation, we have the terms 0.65m and 1.5m. They both have the variable 'm', so they are like terms.
Combining them simplifies our equation. It's just like adding regular numbers:
0.65m + 1.5m = (0.65 + 1.5)m = 2.15m.
Combining like terms makes equations easier to manage and solve.
Isolating the Variable
When solving an equation, our goal is usually to find the value of the variable. To do this, we need to isolate the variable.
This means getting the variable by itself on one side of the equation. In our example, after combining like terms, we have:
2.15m = 0.43.
To isolate 'm', we need to remove the coefficient (2.15) affecting it. This will often involve division, which we'll cover soon.
Simplifying Coefficients
Simplifying coefficients helps make the equation easier to solve. A coefficient is the number in front of a variable.
In our example, once we've combined like terms, we have the simplified coefficient 2.15 for 'm'.
Knowing this coefficient lets us perform the next step in solving our equation.
Simply put, adding or subtracting the coefficients helps reduce the problem to familiar arithmetic.
Division in Algebra
Division is a key operation in isolating the variable. To find the value of the variable 'm', we divide both sides of the equation by the coefficient.
From our equation 2.15m = 0.43, we divide both sides by 2.15:
\[ m = \frac{0.43}{2.15} \]
Performing this division gives us the value of 'm'.
This step allows us to reach the solution:
\[ m = 0.2 \]
Division in algebra is a powerful tool for solving equations and finding unknown values.

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Most popular questions from this chapter

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{6}{35 a^{2}}+\frac{1}{30 a^{3}}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Manuel has scores of \(77,82,89,89,\) and 92 on five class exams. His final exam is going to count double. What score must he get on his final exam so that his total average is at least \(87 ?\)

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{12}{7} \cdot \frac{5}{8}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. The owner of a gourmet coffee shop wants to create 200 pounds of a special blend of coffee to sell at \(\$ 8.15\) per pound. He also wants to use all 80 pounds of the \(\$ 6.50\) blend he has on hand. How expensive should the coffee be that he uses for the remaining 120 pounds?

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Sal has \(\$ 3200\) invested in an account paying \(7.2 \%\) interest per year. At what rate must he invest an additional \(\$ 2800\) so that the annual interest from the two investments will be \(\$ 390 ?\)
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