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Chapter 2: Problem 58

For exercises 1-72, (a) solve. (b) check. $$ 0.3 x-1.8=7.2 $$

Short Answer

Expert verified
x = 30

Step by step solution

01

- Isolate the variable term

Add 1.8 to both sides of the equation to isolate the term containing the variable: \[ 0.3x - 1.8 + 1.8 = 7.2 + 1.8 \] which simplifies to \[ 0.3x = 9.0 \]
02

- Solve for the variable

Divide both sides by 0.3 to solve for the variable: \[ \frac{0.3x}{0.3} = \frac{9.0}{0.3} \] which simplifies to \[ x = 30 \]
03

- Check the solution

Substitute the value of x back into the original equation to verify: \[ 0.3(30) - 1.8 = 7.2 \] Calculating the left side gives \[ 9.0 - 1.8 = 7.2 \] which confirms that the solution is correct.

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Key Concepts

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

Isolating Variables
When solving a linear equation, one of the most crucial steps is isolating the variable. This means we want to get the variable (often labeled as x) on one side of the equation by itself. To achieve this, we perform operations that help simplify the equation.
In the given exercise, the first step was to add 1.8 to both sides. By doing this, we eliminate the constants on the same side as x.
For instance, with the equation:
\[ 0.3 x - 1.8 = 7.2 \] When we add 1.8 to both sides:
\[ 0.3 x - 1.8 + 1.8 = 7.2 + 1.8 \] It simplifies to:
\[ 0.3 x = 9.0 \] This helps us see the term with x more clearly.
Checking Solutions
After finding a potential solution to an equation, it鈥檚 always a good practice to check if that solution is correct.
To check the solution, substitute the value back into the original equation and ensure both sides of the equation equal.
For example, if we found x to be 30 in the following equation:
\[ 0.3 x - 1.8 = 7.2 \] We substitute 30 for x:
\[ 0.3 (30) - 1.8 \] Which becomes:
\[ 9.0 - 1.8 = 7.2 \] Since both sides of the equation are equal, our solution of x = 30 is confirmed. Checking solutions is necessary to avoid errors and ensure our calculations are accurate.
Basic Algebra
Basic Algebra involves understanding and manipulating algebraic expressions and equations. It starts with operations like addition, subtraction, multiplication, and division but applied within algebraic contexts.
In the exercise provided, simple algebraic operations are used:
  • Adding 1.8 to both sides to simplify the equation
  • Dividing both sides by 0.3 to solve for x
Dividing by the coefficient of x (0.3 in this case) is a key algebraic step to isolate and solve for the variable:
\[ \frac{0.3 x}{0.3} = \frac{9.0}{0.3} \] Simplifying to:
\[ x = 30 \] Mastery of these basic algebra concepts is essential for solving more complex equations later on.

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