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Chapter 2: Problem 355

Solve each equation with decimal coefficients. $$0.4 y-4=2$$

Short Answer

Expert verified
The solution is \( y = 15 \).

Step by step solution

01

Isolate the Variable Term

To start solving the equation, isolate the term involving the variable on one side of the equation. For this equation, add 4 to both sides:\(0.4y - 4 + 4 = 2 + 4\)Simplifying this, we get:\(0.4y = 6\)
02

Solve for the Variable

Next, solve for the variable by dividing both sides by the decimal coefficient of the variable term:\(y = \frac{6}{0.4}\)Simplify this by multiplying numerator and denominator by 10 to get:\(y = \frac{60}{4} = 15\)

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Key Concepts

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

Isolating the Variable
When solving algebraic equations, the first step is often to isolate the variable. This means getting the variable by itself on one side of the equation.
In our example equation, \(0.4y - 4 = 2\), we need to move the constant term (\(-4\)) to the other side.
We do this by adding the same value (4) to both sides of the equation: \(0.4y - 4 + 4 = 2 + 4\).
This results in: \(0.4y = 6\)

Always remember that whatever you do to one side of the equation, you must do to the other side.
This keeps the equation balanced and helps isolate the variable efficiently.
Dividing by Decimal Coefficients
Once the variable is isolated, we often need to divide by the coefficient of the variable term to solve for the variable.
A coefficient is a number that multiplies a variable. In our example, the coefficient is 0.4.
To solve for \( y \), divide both sides of the equation by 0.4: \( y = \frac{6}{0.4}\).

Working with decimals can be tricky, so sometimes it's easier to multiply the numerator and denominator by 10 to eliminate the decimal: \( y = \frac{6 \times 10}{0.4 \times 10} = \frac{60}{4}\).
Finally, divide 60 by 4 to get \( y = 15 \).

Dividing by decimal coefficients might seem complicated at first, but simplifying using multiplication helps clarify the process.
Algebraic Equations
Algebraic equations are mathematical statements where two expressions are set equal to each other.
These expressions typically include variables, numbers, and operators like addition, subtraction, multiplication, and division.
The goal in solving an algebraic equation is to find the value of the variable that makes the equation true.
Here are a few tips for solving these equations:
  • Isolate the variable term on one side of the equation.
  • Use inverse operations (addition vs. subtraction, multiplication vs. division) to simplify the equation.
  • Double-check your work by substituting the solution back into the original equation.

Solving algebraic equations builds a foundation for understanding more complex math concepts. Practice these techniques, and soon they will become second nature.

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Most popular questions from this chapter

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$\frac{9}{4} g \leq 36$$

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