Question

Write equivalent compound statements if possible. a. x = 2 *x b. x = x + y - 2; c. sum = sum + num; d. z = z *x + 2 *z; e. y = y / (x + 5);

Short answer

a. x = 0; b. y = 2; c. num = 0; d. z = 0 or x = -1; e. x = -4.

Step-by-step solution

Simplifying equation a

The equation given is \( x = 2 \times x \). This represents a simple multiplication equality. To simplify it, subtract \( x \) from both sides: \( x - x = 2x - x \), resulting in \( 0 = x \). This indicates that the equation holds true when \( x = 0 \).

Simplifying equation b

The equation is \( x = x + y - 2 \). Subtract \( x \) from both sides to isolate the variable: \( x - x = x + y - 2 - x \), resulting in \( 0 = y - 2 \). This indicates \( y = 2 \) as the equivalent statement.

Simplifying equation c

The equation is \( \text{sum} = \text{sum} + \text{num} \). Subtract \( \text{sum} \) from both sides: \( \text{sum} - \text{sum} = \text{sum} + \text{num} - \text{sum} \), resulting in \( 0 = \text{num} \). This implies the equivalent statement \( \text{num} = 0 \).

Simplifying equation d

The equation is \( z = z \times x + 2 \times z \). Factor out the common term \( z \) from the right side: \( z = z (x + 2) \). For this equation to be equivalent, either \( z = 0 \) or \( x + 2 = 1 \). This results in \( z = 0 \) or \( x = -1 \) as possible equivalent statements.

Simplifying equation e

The equation is \( y = \frac{y}{x + 5} \). Multiply both sides by \( x + 5 \) to eliminate the fraction: \( y(x + 5) = y \). Simplify by dividing both sides by \( y \) (assuming \( y eq 0 \)): \( x + 5 = 1 \). This results in the equivalent statement \( x = -4 \).

Key concepts

Simplifying Equations

Simplifying equations involves reducing an expression into a simpler or more manageable form while retaining its original meaning. This process is important to make mathematical problems easier to solve or analyze. Let's look at some key approaches used when simplifying equations, based on our initial examples.

Firstly, subtraction of terms from both sides of an equation is a common technique. For example, if you have the equation \( x = 2 \times x \), you can subtract \( x \) from both sides to simplify it to \( 0 = x \). By doing this, you are eliminating the common variable on both sides, resulting in a simpler expression showing that \( x = 0 \).

Another technique is to factor common terms, which can be seen in the equation \( z = z \times x + 2 \times z \). By factoring out \( z \), it becomes \( z = z(x+2) \), making it clear that either \( z = 0 \) or the expression in the parenthesis has to equate to 1. This simplification clarifies the potential solutions like \( z = 0 \) or \( x = -1 \).

Simplifying fractions by eliminating denominators or finding equalities between expressions are other methods. In the equation \( y = \frac{y}{x + 5} \), multiplying through by \( x+5 \) eliminates the fraction, simplifying it to \( x + 5 = 1 \) and revealing that \( x = -4 \). Through these techniques, solving equations becomes less complex and more focused on deriving meaningful mathematical constraints.

Problem-Solving Steps

Understanding and systematically applying problem-solving steps can greatly aid in tackling math exercises. Structuring your approach can lead to clarity and effective solutions.

Start with analyzing the given equation. Take note of all terms involved and observe any operations occurring between them. This initial understanding is crucial, as seen in the transformation of \( x = x + y - 2 \) where isolating each term helps recognize that \( y = 2 \).

Next, perform appropriate algebraic operations to manipulate the equation into a simpler form. This often involves the use of addition, subtraction, multiplication, or division to both sides. For instance, subtracting \( x \) from both sides helps in simplifying equation \( x = x + y - 2 \) to \( 0 = y - 2 \).

After manipulation, evaluate the equation to identify constraints or specific solutions, like when we deduce \( num = 0 \) from \( \text{sum} = \text{sum} + \text{num} \) by balancing both sides of the equation.

• Analyze and identify all variables and terms.
• Subtract or add terms to simplify.
• Factor, divide or multiply to get equalities.
• Check constraints or specific solutions.

Following these steps not only helps in simplifying the expressions but also guides in finding equivalent or meaningful statements that can be directly interpreted.

Algebraic Manipulation

Algebraic manipulation is the backbone of solving complex equations in mathematics, allowing us to transform equations and express them in a more useful way. Let's delve into how this applies to our equations.

To carry out successful algebraic manipulation, you must be comfortable with manipulating each side of an equation using operations that maintain equality. For instance, in the equation \( sum = sum + num \), subtracting \( sum \) from both sides allows us to isolate \( num \), leading to the realization that \( num = 0 \).

Sometimes, factorization is necessary when faced with complex expressions as seen in \( z = z * x + 2 * z \). By pulling out \( z \) as a common factor, it simplifies to \( z(x+2) = z \). This significantly reduces complexity and allows us to see that either \( z = 0 \) or \( x = -1 \) holds true separately.

Another key manipulative skill is balancing equations by maintaining equivalent operations. This can involve multiplication or division across the entire equation, as demonstrated in \( y = \frac{y}{x+5} \). Multiplying each term by \( x+5 \) renders the fraction unnecessary, simplifying the equation to its core components of \( x= -4 \).

These techniques and understanding of algebraic manipulations enable us to target core components of problems directly, making equations much simpler to handle.