Problem 39 - Algebra A Combined Function

Question

Write equivalent equations by multiplying both sides of each given equation by the given nonzero number. $$ -4 x+y=3 \text { by } 3 $$

Step-by-Step Solution

Answer

The equivalent equation is \(-12x + 3y = 9\).

Step 1: Understanding Equivalent Equations

An equivalent equation is one that has the same solution as the original. We can obtain an equivalent equation by performing the same operations on both sides of the equation without changing its solution.

Step 2: Identifying the Equation and Multiplier

The given equation is \(-4x + y = 3\) and the multiplier is 3. This means we need to multiply every term in the equation by 3 to get an equivalent equation.

Step 3: Multiplying Both Sides of the Equation

Multiply each term in the equation by 3. This gives us: \(3(-4x) + 3(y) = 3(3)\), resulting in \(-12x + 3y = 9\).

Step 4: Writing the Equivalent Equation

The equivalent equation after multiplying each term by 3 is \(-12x + 3y = 9\). This equation represents the same relationship between \(x\) and \(y\) as the original equation.

Key Concepts

Multiplying Equations

When we talk about multiplying equations, we refer to the process of taking every term in an equation and multiplying it by the same non-zero number. This is a crucial algebraic technique because it allows you to create equivalent equations without altering the original equation's solution.
To understand why this works, consider that an equation is like a balanced scale; multiplying both sides by the same quantity maintains that balance.

• For example, if you have the equation \(-4x + y = 3\), multiplying each term by 3 gives \(-12x + 3y = 9\).
• Every term must be multiplied, including both variables and constants.

This procedure is commonly used to eliminate fractions or clear denominators, simplifying solving the equation.

Algebraic Manipulation

Algebraic manipulation involves changing the form of an equation or inequality using valid mathematical operations while keeping its solutions intact. This concept encompasses a wide range of techniques, including addition, subtraction, multiplication, division, and the use of properties such as the distributive property.
One common use of algebraic manipulation is finding equivalent equations.

• By multiplying the original equation's terms by a specific non-zero number, you transform it into a more usable form.
• This process can help in solving systems of equations or in simplifying complex expressions.

Knowing how and when to apply these manipulations is key to mastering algebra.

Solutions to Equations

Solutions to equations are the specific values of the variables that satisfy the equation. In other words, they make the equation true. Understanding how to obtain and interpret these solutions is essential in algebra and beyond.
Equivalent equations will always have the same set of solutions since they represent the same mathematical relationship.

• Consider that the solutions to \(-4x + y = 3\) are the same as to \(-12x + 3y = 9\) after multiplying by 3.
• This is because the fundamental relationship between the variables remains unchanged.

Understanding this principle helps in solving equations and verifying your results. It ensures that even if the appearance of the equation changes, its core relationships stay the same.