Solve for x -2-8(5-4x)=36-8x

Solution:

Step 1: Simplify the expression on the left-hand side.

Step 1.1: Break down the expression term by term.

Step 1.1.1: Use the distributive property to expand $-8(5 - 4x)$.

$-2 - 8 \cdot 5 + 8 \cdot 4x = 36 - 8x$

Step 1.1.2: Perform the multiplication of $-8$ and $5$.

$-2 - 40 + 8 \cdot 4x = 36 - 8x$

Step 1.1.3: Multiply $8$ and $-4x$.

$-2 - 40 + 32x = 36 - 8x$

Step 1.2: Combine like terms on the left-hand side.

$-42 + 32x = 36 - 8x$

Step 2: Gather all the $x$ terms on one side of the equation.

Step 2.1: Add $8x$ to both sides to move the $x$ terms to the left.

$-42 + 32x + 8x = 36$

Step 2.2: Combine the $x$ terms.

$-42 + 40x = 36$

Step 3: Isolate the constant terms on the opposite side of the $x$ terms.

Step 3.1: Add $42$ to both sides to move constants to the right.

$40x = 36 + 42$

Step 3.2: Add the numbers on the right-hand side.

$40x = 78$

Step 4: Solve for $x$ by dividing both sides by the coefficient of $x$.

Step 4.1: Divide the equation by $40$.

$\frac{40x}{40} = \frac{78}{40}$

Step 4.2: Simplify the equation.

Step 4.2.1: Reduce the left-hand side.

$\frac{\cancel{40}x}{\cancel{40}} = \frac{78}{40}$

Step 4.2.2: Simplify to find $x$.

$x = \frac{78}{40}$

Step 4.3: Further simplify the fraction.

Step 4.3.1: Find and cancel common factors.

$x = \frac{2 \cdot 39}{2 \cdot 20}$

Step 4.3.2: Reduce the fraction by canceling common factors.

$x = \frac{\cancel{2} \cdot 39}{\cancel{2} \cdot 20}$

Step 4.3.3: Write the simplified fraction.

$x = \frac{39}{20}$

Step 5: Present the solution in various formats.

Exact Form: $x = \frac{39}{20}$ Decimal Form: $x = 1.95$ Mixed Number Form: $x = 1 \frac{19}{20}$

Knowledge Notes:

The problem-solving process involves several key algebraic techniques:

1. Distributive Property: This property allows us to multiply a single term by each term inside a parenthesis, e.g., $a(b + c) = ab + ac$.

2. Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power, e.g., $5x + 3x = 8x$.

3. Isolating the Variable: To solve for a variable, we aim to get the variable on one side of the equation and the constants on the other, often by using addition or subtraction.

4. Solving Linear Equations: Once the variable is isolated, we can solve the equation by performing operations that simplify the equation, such as dividing both sides by the coefficient of the variable.

5. Simplifying Fractions: This involves finding the greatest common factor of the numerator and denominator and dividing both by that number to reduce the fraction to its simplest form.

6. Multiple Representations of Numbers: A number can be represented in various forms, including exact fraction form, decimal form, and mixed number form. It is important to be able to convert between these forms.