Problem 22 - Prealgebra

Question

Solve each equation. $$0.1 x+0.5(x+8)=7$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Distribute 0.5

We're given the equation: \[0.1 x + 0.5(x + 8) = 7\]Distribute the 0.5 to both \(x\) and 8. This changes the equation to:\[0.1 x + 0.5 imes x + 0.5 imes 8 = 7\]Simplify to:\[0.1 x + 0.5x + 4 = 7\]

Step 2: Combine Like Terms

Combine the like terms \(0.1x\) and \(0.5x\):\[0.1x + 0.5x + 4 = 7\]This simplifies to:\[0.6x + 4 = 7\]

Step 3: Isolate the Variable

Subtract 4 from both sides to isolate the term with \(x\):\[0.6x + 4 - 4 = 7 - 4\]This results in:\[0.6x = 3\]

Step 4: Solve for x

Divide both sides by 0.6 to solve for \(x\):\[x = \frac{3}{0.6}\]Simplify the fraction by converting it to a whole number:\[x = \frac{30}{6} = 5\]

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept when it comes to solving equations. It allows us to simplify expressions in which a term is multiplied by terms inside a parenthesis. To apply the distributive property, you multiply the term outside the parenthesis by each term inside. For example, in our equation, we have: - The term 0.5 is outside the parenthesis, which contains 'x + 8'. - Applying the distributive property, we multiply 0.5 by both 'x' and 8 separately. This transforms the expression 0.5(x + 8) into 0.5x + 0.5 * 8, which simplifies to 0.5x + 4. Remember:

• The distributive property is helpful for removing parentheses.
• It is essential for re-writing the equation in a simpler form that is easier to work with.

Use this step whenever you encounter parentheses in an equation with a term being multiplied by the entire expression inside.

Mastering Combining Like Terms

When solving equations, combining like terms helps in simplification by consolidating similar variables and constants. Like terms in an equation are terms that contain the same variable raised to the same power or are constants. In our equation: - After applying the distributive property, we have 0.1x + 0.5x. - These are like terms because they both have the variable 'x'. You add the coefficients of these like terms to simplify them: 0.1x + 0.5x becomes 0.6x. Key Points:

• Combine coefficients of terms with the same variable to make mathematics simpler and easier to handle.
• Always check for additional like terms throughout the equation after each simplification step.

By combining like terms, you simplify the equation, making it much easier to isolate the variable and ultimately solve for it.

Techniques for Isolating the Variable

Isolating the variable is the process of manipulating the equation to get the variable you're solving for on one side, usually the left, by itself. This is crucial for finding the solution to the equation. Here's how to isolate the variable in our example: - After combining like terms, your equation is 0.6x + 4 = 7. - Begin by eliminating the constant term from the side of the equation that contains the variable. Subtract 4 from both sides to get 0.6x = 3. Next: - To completely isolate x, divide both sides by the coefficient 0.6. This gives you x = 3 / 0.6. - Simplifying the fraction gives x = 5. Tips:

• Always perform the same operation on both sides of the equation to maintain equality.
• In cases involving fractions or decimals, simplify them to make calculations easier.

By successfully isolating the variable, you make the equation solvable, revealing the value of the unknown in its simplest form.