Problem 5 - Elementary Algebra

Question

For Problems 1-12, solve each equation. You will be using these types of equations in Problems \(13-41\). $$ 0.7(15)-x=0.6(15-x) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3.75\).

Step 1: Distribute the Constants

First, apply distribution on both sides of the equation. Multiply 0.7 by 15 on the left side and 0.6 by both 15 and \(-x\) on the right side.So we have:\[0.7 \times 15 - x = 0.6 \times 15 - 0.6x\]

Step 2: Simplify the Equation

Now, simplify each side by performing the multiplication:\[10.5 - x = 9 - 0.6x\]

Step 3: Rearrange Terms

In this step, move the variable terms to one side and constant terms to the other. Let's add \(x\) to both sides first:\[10.5 = 9 + 0.4x\]

Step 4: Isolate the Variable

Subtract 9 from both sides to isolate the term with \(x\):\[1.5 = 0.4x\]

Step 5: Solve for x

Divide both sides by 0.4 to solve for \(x\):\[x = \frac{1.5}{0.4}\]Calculate the division:\[x = 3.75\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that simplifies how we deal with multiplication over addition or subtraction. This property allows you to multiply a number by a group of numbers added or subtracted together. In mathematical terms, it means:

• If you have an expression like \(a(b+c)\), you can distribute \(a\) and rewrite it as \(ab + ac\).
• This rule doesn’t only apply to addition but also to subtraction, so \(a(b-c) = ab - ac\).

In the provided exercise, we applied this rule to break down:

• \(0.7(15)\) on the left side of the equation, resulting in \(0.7 \times 15\)
• \(0.6(15 - x)\) on the right side, leading to two separate terms: \(0.6 \times 15\) and \(-0.6x\)

Using the distributive property helps simplify the problem into smaller, more manageable pieces. This makes it easier to isolate variables or combine like terms later on.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is a mathematical sentence of the form \(ax + b = c\), where \(x\) is the variable. Here is a basic approach to solving such equations:

• Start by simplifying both sides of the equation if needed. This could involve distributing multiplication or combining like terms.
• Move all terms involving the variable to one side of the equation, using addition or subtraction.
• Move constant terms to the opposite side, again using addition or subtraction.

In our example, after using the distributive property, the equation \(10.5 - x = 9 - 0.6x\) was simplified. To solve for \(x\), we rearranged terms so that:

• We add \(x\) to both sides to bring variable terms together.
• This step resulted in \(10.5 = 9 + 0.4x\).

The goal is to isolate the variable on one side, which leads us into the next concept.

Variable Isolation

Variable isolation is a key part of solving equations, where you manipulate the equation until the variable is alone on one side. You achieve this through addition, subtraction, multiplication, or division, depending on the equation structure. To isolate the variable:

• Move all constant terms to the opposite side of the variable. In our exercise, subtracting 9 from both sides gave \(1.5 = 0.4x\).
• Next, divide or multiply to solve for \(x\). Here, dividing both sides by 0.4 isolated \(x\), resulting in \(x = 3.75\).

Isolating the variable allows you to discover its specific value. This process ensures precision in finding your solution to the equation.