Problem 18 - Elementary and Intermediate Algebra

Question

The expression \(4-y\) must be multiplied by _____ to obtain \(y-4\).

Step-by-Step Solution

Answer

Multiply \(4-y\) by \(-1\) to get \(y-4\).

Step 1: Understanding the Problem

We have the expression \(4-y\) and we want to find a number that we can multiply it by to obtain \(y-4\). The task is to determine this number.

Step 2: Analyze the Expressions

Notice that \(y-4\) is the opposite of \(4-y\). In general, if you want to get the opposite expression, you can multiply by \(-1\).

Step 3: Test the Hypothesis

Multiply \(4-y\) by \(-1\): \(-1 \cdot (4-y) = -1(4) + -1(-y) = -4 + y\), which simplifies to \(y-4\).

Step 4: Confirm the Solution

The multiplication \(-1 \cdot (4-y) = y-4\) confirms that multiplying by \(-1\) flips the signs of each term, turning \(4-y\) into \(y-4\). Hence, the expression \(4-y\) must be multiplied by \(-1\) to obtain \(y-4\).

Key Concepts

Expressions

An algebraic expression is a combination of terms that are combined using mathematical operators like addition, subtraction, multiplication, or division. In the given problem, we have expressions like \(4-y\) and \(y-4\). An expression with two terms, such as these, is often called a binomial.
Understanding expressions is fundamental in algebra since they allow us to represent mathematical situations or concepts succinctly. In our example, we want to change one expression into another by multiplication.
Key things to remember about expressions:

• They can consist of variables, constants, and operators.
• They represent a value but don't always equate to something specific without further definitions or operations.
• Expressions can be simplified, expanded, or manipulated using algebraic laws.

Once you're comfortable with expressions, solving algebraic problems, like the one we're tackling, becomes much simpler.

Multiplication

Multiplication is one of the core mathematical operations, and it plays a vital role in transforming expressions. When multiplying expressions, each term in an expression must be multiplied by the external factor or another term in another expression, following the distributive property.
In our problem, we're using multiplication to transform the expression \(4-y\) into \(y-4\). This involves using multiplication by \(-1\).
Here's what happens during multiplication:

• Each term in the expression is multiplied by the number outside the parentheses.
• Multiplication by a negative number reverses the sign of the terms being multiplied.
• We distribute the multiply effect into every term inside the parentheses.

For example, when we multiply \(4-y\) by \(-1\), we apply the effect to both \(4\) and \(y\), resulting in their signs flipping. This is how multiplication helps us turn \(4-y\) into \(y-4\).

Negative Numbers

Negative numbers are numbers less than zero and carry a "negative" or "minus" sign. They are crucial in algebra, especially when dealing with subtraction or reversing operations using tools like multiplication.
Negative numbers have distinct rules:

• Multiplying two negative numbers results in a positive product.
• Multiplying a positive and a negative number results in a negative product.
• Switching signs in expressions or equations can be easily achieved by multiplying by \(-1\).

In the provided exercise, the operation of multiplying the expression \(4-y\) by \(-1\) transforms it to \(y-4\). This shows us how negative numbers, particularly multiplying by \(-1\), effectively change the signs of all terms in an expression, achieving the opposite but equivalent form of the original expression.