Problem 46 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(-14 y-1.8=-24 y+3.9\)

Step-by-Step Solution

Answer

The solution to the equation is \(y = 0.57\).

Step 1: Isolate all terms with variables on one side

Start by adding or subtracting terms with variables to get them on one side of the equation. Here, add \(24y\) to both sides to move the \(y\) terms together:\[-14y - 1.8 + 24y = -24y + 3.9 + 24y\]This simplifies to:\[10y - 1.8 = 3.9\]

Step 2: Isolate the variable term

Next, get the \(y\) term by itself by adding \(1.8\) to both sides:\[10y - 1.8 + 1.8 = 3.9 + 1.8\]This simplifies to:\[10y = 5.7\]

Step 3: Solve for the variable

Divide both sides by 10 to solve for \(y\):\[y = \frac{5.7}{10}\]Simplify the fraction:\[y = 0.57\]

Key Concepts

Linear Equations

Linear equations might be among the first concepts you'll encounter in algebra. They are called "linear" because they graph as straight lines. Think of them as equations like "y = mx + b" where you'll often seek the value of a variable, like "y." In our given equation,

• The goal is solving for "y." The equation stems from setting two linear expressions equal.
• Each side of the equation can be seen as a separate line if it were graphed.
• Unlike more complex equations, linear equations do not include powers or exponents.

To solve them, we find the value that makes the equation true. In other words, we are on a mission to determine what number "y" must be, such that the left side equals the right side when we substitute "y" in. Linear equations are foundational to understanding algebra and mastering them is a step toward tackling even more complex mathematical relationships.

Variable Isolation

Variable isolation is a technique used to solve equations by getting the variable on its own. In this problem, the variable we need to focus on is "y." The steps are like untangling a set of headphones - methodical but straightforward.

• First, we group all the terms with "y" to one side of the equation. To do this, we add or subtract similar terms from both sides.
• Next, any other numbers, or constants, are moved to the opposite side. This is achieved using the inverse operations such as addition and subtraction.

In our example with \(-14y - 1.8 = -24y + 3.9\), we wanted to go from "scattered" terms to concise expression:

• We actually got the "y" terms all on one side with \(10y = 3.9 + 1.8\).
• Ensuring no y's were left out was important to isolating our focus purely on solving.

This way of isolating makes problem-solving systematic and clearer!

Simplifying Fractions

Simplifying fractions is the process of making them as simple, or as little, as possible. When an equation involves fractions, it often means division is involved at some point.

• For example, when solving for \(y\), we simplified the fraction \(\frac{5.7}{10}\).
• The aim here is to reduce it to its simplest form or least complex numeric expression, which is \(y = 0.57\) in this case.
• By doing this, the answer becomes easier to digest and verify.

Simplifying makes calculations straightforward and number handling less prone to error. It's always a handy tool in your math toolkit, ensuring the solutions are clear and comprehensible.