Problem 14 - Algebra for College Students

Question

Solve each equation. \(0.07 x=152-0.08(2000-x)\)

Step-by-Step Solution

Answer

The solution is \( x = 800 \).

Step 1: Distribute the negative sign

We need to distribute the \( -0.08 \) across the terms inside the parentheses on the right side of the equation: \( 0.07x = 152 - 0.08 imes 2000 + 0.08x \)

Step 2: Compute the multiplication

Calculate \( -0.08 imes 2000 = -160 \). The equation now becomes:\( 0.07x = 152 - 160 + 0.08x \)

Step 3: Combine like terms on the right

Combine the constants on the right side: \( 0.07x = -8 + 0.08x \)

Step 4: Move the variable to one side

Subtract \( 0.08x \) from both sides to get all terms involving \( x \) on one side of the equation:\( 0.07x - 0.08x = -8 \)

Step 5: Simplify the equation

Combine the \( x \) terms: \( -0.01x = -8 \)

Step 6: Solve for x

Divide both sides by \( -0.01 \) to solve for \( x \):\( x = \frac{-8}{-0.01} = 800 \)

Key Concepts

Distributive Property

The distributive property is a crucial algebraic principle used to simplify expressions and solve equations. It involves distributing, or multiplying, a single term across terms within parentheses. The generic form of the distributive property is:
\( a(b + c) = ab + ac \). This means if you have a term outside the parenthesis, you multiply it with each term inside the parentheses.
In our example, we used this property to distribute the \( -0.08 \) across the terms in the expression

• \( -0.08 \times 2000 = -160 \)
• \( -0.08 \times (-x) = 0.08x \)

Understanding how to properly apply the distributive property is key in moving forward with algebra problems, as it allows cases where we turn more complex multiplications into easier steps by breaking them down.

Combining Like Terms

Combining like terms simplifies mathematical expressions and equations. Like terms are terms that contain the same variable raised to the same power. For example, in the expression \( 3x + 5 + 2x -4 \), the like terms \( 3x \) and \( 2x \) can be combined to become \( 5x \), while the constants \( 5 \) and \( -4 \) become 1.
In the provided exercise, after applying the distributive property, we had the equation

• \( 0.07x = 152 - 160 + 0.08x \).
  This was further simplified by combining the constant terms \( 152 \) and \( -160 \), which resulted in \( -8 \).
  This gives us the simplified equation of:
• \( 0.07x = -8 + 0.08x \).

Efficiently combining like terms reduces the complexity of equations, making them easier to solve.

Isolation of Variables

Isolation of variables is a fundamental step when solving linear equations. The goal is to get the variable, typically denoted as \( x \), by itself on one side of the equation. This means rearranging the equation so that \( x \) is isolated, and all the other numbers are on the opposite side.
In the original problem, we started with
\( 0.07x = -8 + 0.08x \).
To isolate \( x \), we moved terms by subtracting \( 0.08x \) from both sides. This results in:
\( 0.07x - 0.08x = -8 \).
This simplifies to:

• \( -0.01x = -8 \)

Finally, we divided both sides by \( -0.01 \) to completely isolate \( x \):

• \( x = \frac{-8}{-0.01} = 800 \)

Once you have the variable isolated and found the value, you've completed the equation solving process!