Problem 40 - Algebra A Combined Function

Question

Write equivalent equations by multiplying both sides of each given equation by the given nonzero number. $$ 5 a-7 b=-4 \text { by }-4 $$

Step-by-Step Solution

Answer

The equivalent equation is \(-20a + 28b = 16\).

Step 1: Identify the Equation and the Multiplier

We are given the equation \(5a - 7b = -4\) and need to multiply both sides by the nonzero number \(-4\).

Step 2: Multiply Both Sides by the Multiplier

Multiply each term of the equation \(5a - 7b = -4\) by \(-4\): \[ -4(5a - 7b) = -4(-4) \] Start with the left side: \[ -4 \cdot 5a - 4 \cdot (-7b) \] And the right side becomes: \[ -4 \cdot (-4) \]

Step 3: Simplify the Equation

Simplify the multiplication: For the left side:\[ -20a + 28b \] For the right side:\[ 16 \] So the equation now is: \[-20a + 28b = 16\]

Key Concepts

Algebraic Equations

Algebraic equations are statements that express an equality involving algebraic expressions. In simpler terms, they are mathematical sentences that say two things are equal and involve variables like \(a\) and \(b\) in the expression. In an algebraic equation, variables are used to represent numbers, allowing us to write general rules about numbers. When working with algebraic equations, the goal is often to find the value of the variables that make the equation true, known as solving the equation. Understanding the balance between both sides of an equation is crucial, as one side must always equal the other for the equation to hold true. This principle of balance is what allows us to manipulate and solve these equations.

Multiplication in Algebra

Multiplication in algebra is similar to arithmetic multiplication but includes variables and sometimes constants. It is a key operation used to transform equations into a new form. When performing algebraic multiplication, each term in the equation is multiplied individually. For instance, if given an equation such as \(5a - 7b = -4\), and asked to multiply by -4, you'd apply the multiplication to each term separately. Pay special attention to the signs - multiplying two negative numbers results in a positive number. The distributive property, \(-4(5a - 7b) = -4 \cdot 5a + (-4) \cdot (-7b)\), is a useful tool here. It helps ensure that each term is appropriately accounted for.

Simplifying Equations

Simplifying equations involves breaking down the equation to its most basic form while ensuring that both sides remain equal. It often includes performing basic operations such as multiplication, combining like terms, and reducing expressions. In the given exercise, after multiplication, we combined the terms \(-20a\) and \(28b\) on one side, with \(16\) on the other side. Simplification helps make complex equations more manageable and is essential for the process of solving equations. Remember, while simplification changes the equation's appearance, it doesn't change the relationship among the variables or constants involved. It maintains the equation's integrity, keeping it equivalent to its original form.