Problem 37 - Algebra A Combined Function

Question

Write equivalent equations by multiplying both sides of each given equation by the given nonzero number. $$ 3 x+2 y=6 \text { by }-2 $$

Step-by-Step Solution

Answer

The equivalent equation is \(-6x - 4y = -12\).

Step 1: Identify the Given Equation and Number

The given equation is \(3x + 2y = 6\) and the nonzero number to multiply by is \(-2\). Our first task is to understand that we need to apply this number to every term in the equation.

Step 2: Apply Multiplication to the Left Side of the Equation

Multiply each term on the left side of the equation by \(-2\):\(-2 \times 3x + (-2) \times 2y\). This simplifies to \(-6x - 4y\).

Step 3: Apply Multiplication to the Right Side of the Equation

Multiply the right side of the equation, \(6\), by \(-2\):\(-2 \times 6\). This simplifies to \(-12\).

Step 4: Write the Equivalent Equation

Combine the results from Steps 2 and 3 to form the new equation. This results in \(-6x - 4y = -12\).

Key Concepts

Multiplication of Equations

When we talk about multiplication of equations, we're referring to multiplying every term in an equation by the same nonzero number. This process is crucial to maintaining the equality between both sides of the equation. Suppose you have an equation like \(3x + 2y = 6\). Here, multiplying each term by a number, say \(-2\), means you need to multiply both the left side terms (i.e., \(3x\) and \(2y\)) and the right side term (i.e., \(6\)) by \(-2\). This is done step by step:

• First, multiply the term \(3x\) by \(-2\) to get \(-6x\).
• Then, multiply the term \(2y\) by \(-2\) to get \(-4y\).
• Finally, multiply \(6\) by \(-2\) to get \(-12\).

This operation is fundamental and must be applied to each part of the equation to ensure accuracy. This technique comes handy in algebra for solving and transforming equations into a more usable form.

Equivalent Equations

Equivalent equations are equations that have the same solution set. They may look different on the outside, but under the hood, they produce the same results when solved. When you multiply every term of an equation by the same nonzero number, you are creating an equivalent equation.
For example, transforming \(3x + 2y = 6\) into \(-6x - 4y = -12\) by multiplying everything by \(-2\) is a classic way of obtaining an equivalent equation. Here's why this works:

• Both original and new equations would yield the same values for \(x\) and \(y\) when solved.
• This method is often used to simplify equations for easier handling, especially in solving systems of equations.
• It allows us to manipulate equations without changing their inherent characteristics.

Understanding equivalent equations helps in recognizing how equations can be altered for simplicity without losing their meaning or solution sets.

Algebraic Manipulation

Algebraic manipulation is the art of changing the form of an equation to make it easier to work with. It includes operations such as addition, subtraction, multiplication, and division applied to both sides of an equation to simplify or rearrange it.
In the context of our exercise, multiplying the equation by \(-2\) is a form of algebraic manipulation. This step involves altering both sides of \(3x + 2y = 6\) consistently:

• By multiplying \(3x\) and \(2y\) by \(-2\), you transform the left-hand side to \(-6x - 4y\).
• Similarly, multiply \(6\) to get \(-12\) on the right-hand side.

Such manipulations are crucial when solving equations, as they help isolate variables or form desired expressions for further calculations. Learning these techniques enhances problem-solving skills and makes complex algebraic concepts more manageable.