Problem 22 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ 0.40 x+0.06(30)=9.8 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 20 \).

Step 1: Expand the equation

First, distribute the 0.06 to the 30 inside the parentheses. This will simplify the equation. The equation becomes: \[ 0.40x + 0.06 \times 30 = 9.8 \] Calculate the multiplication: \[ 0.06 \times 30 = 1.8\] So the equation transforms into: \[ 0.40x + 1.8 = 9.8 \]

Step 2: Isolate the variable term

Subtract 1.8 from both sides of the equation to get the term with the variable alone on one side. \[ 0.40x + 1.8 - 1.8 = 9.8 - 1.8 \] This simplifies to: \[ 0.40x = 8.0 \]

Step 3: Solve for the variable

Divide both sides of the equation by 0.40 to solve for \( x \). \[ \frac{0.40x}{0.40} = \frac{8.0}{0.40} \] Calculate the division on the right side: \[ x = 20 \]

Key Concepts

Distributive Property

In the world of algebra, the distributive property is like a handy toolkit for simplifying expressions and equations. When you have an equation with a number outside parentheses, like in our exercise, you use the distributive property to "distribute" that number to everything inside the parentheses.

Here's how it works:

• Take the number outside the parentheses (in our case, 0.06) and multiply it by each term inside the parentheses (30).
• This math process is: \[0.06 \times 30 = 1.8\]
• The equation then changes from \[0.40x + 0.06(30) = 9.8\]to \[0.40x + 1.8 = 9.8\]

Using the distributive property effectively simplifies the equation, allowing us to then focus on just the variable term and constants outside the parentheses. This is the first step in our journey to solve the equation.

Isolating the Variable

Once we've used the distributive property, our main goal becomes isolating the term with the variable. This is an essential skill in solving linear equations because it's all about transforming the equation so that the variable stands on its own on one side of the equation.

Here's the process:

• You want the variable term, in this case, \(0.40x\), to be the only term on one side of the equation.
• We had \[0.40x + 1.8 = 9.8\]
• Subtract 1.8 from both sides to remove it from the left side:\[0.40x + 1.8 - 1.8 = 9.8 - 1.8\]
• This simplifies to:\[0.40x = 8.0\]

By subtracting 1.8 from both sides, we have successfully isolated \(0.40x\), preparing us to easily solve for the variable in the next step. Remember to always perform the same operation on both sides to maintain the balance of the equation.

Solving Equations

The final mission is to solve the equation, which means finding the exact value of the variable. Now that \(0.40x\) is isolated, the final step is straightforward.

The steps to solve for \(x\) are:

• We have the equation \[0.40x = 8.0\]
• We need to divide both sides by 0.40 to solve for \(x\): \[\frac{0.40x}{0.40} = \frac{8.0}{0.40}\]
• This simplifies to:\[x = 20\]

And there we have it! The variable \(x\) equals 20. Always remember that whatever operation you use to isolate the variable, you must apply it equally to both sides of the equation. This keeps the equation balanced, leading you to the correct solution.