Problem 18 - Algebra 1

Question

Write the equation in the form \(a x+b=0\). Then write the related function \(y=a x+b\). $$-4 x-5=3 x+8$$

Step-by-Step Solution

Answer

The equation in the form \(ax + b = 0\) is \(-7x - 13 = 0\) and the related function is \(y = -7x - 13\).

Step 1: Rearrange The Given Equation

Combine similar terms. Our equation is \(-4x - 5 = 3x + 8\). Bring all x terms to one side and numbers to the other side: this gives \(-4x - 3x = 8 + 5\).

Step 2: Simplify The Equation

Simplify both sides of the equation by performing the necessary mathematical operations. We will get \(-7x = 13\). As the equation is expected to be in the form \(a x + b = 0\), further move 13 to the left side which gives \(-7x - 13 = 0\). This is our equation in the form \(ax + b = 0\). Where a=-7 and b=-13.

Step 3: Write The Related Function \(y = ax + b\)

Replace the zero from the equation \(-7x - 13 = 0\) with 'y'. This gives the function \(y = -7x - 13\). This is the related function.

Key Concepts

Rearranging Equations

Rearranging equations is an essential skill that allows you to manipulate mathematical expressions to isolate desired variables or rewrite in standard forms. It involves using algebraic rules and operations to achieve a simplified or specified format.
To rearrange an equation properly:

• Identify terms that contain the variable of interest and move them to one side of the equation.
• The remaining terms should be shifted to the opposite side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides equally.
• Follow mathematical conventions to order the terms, often arranging from the highest degree of the variable to the lowest.

In the provided example, the goal was to rewrite \(-4x - 5 = 3x + 8\) into the form \(ax + b = 0\). Moving all "\(x\)" terms to one side and constants to the other helps in achieving this format by using operations like addition and subtraction. Once rearranged, the equation represented a format where coefficients and constants are isolated, preparing it for further manipulation or analysis like simplifying or solving.

Function Notation

Function notation is a way of expressing a mathematical relationship between variables. In traditional equations, we often see them expressed as \(ax + b = 0\). Function notation introduces the concept of a function, written as \(y = ax + b\), where "\(y\)" represents the output or the function value for any given "\(x\)" input.
This notation is extremely useful because:

• It highlights the dependence of one variable on another.
• It provides a clear framework to evaluate the function at specific points.
• It's helpful in graphing the linear relationship between "\(x\)" and "\(y\)".

In the example provided, converting \(-7x - 13 = 0\) into a function notation by switching to \(y = -7x - 13\) indicates that for every input value "\(x\)", the output is calculated using \(-7x - 13\). In short, function notation provides a clear format to not only represent an equation but also to delve into its relations and behaviors.

Simplifying Expressions

Simplifying expressions involves reducing equations to their most basic form using basic arithmetic operations and algebraic principles. This process makes expressions easier to work with, clearer to interpret, and quicker to evaluate.
Simplifying typically includes:

• Combining like terms, which refers to grouping and simplifying terms with the same variable raised to the same power.
• Performing basic arithmetic to merge constants or simple values.
• Applying distributive properties when necessary to eliminate parentheses.

In the context of the given exercise, this can be seen in the step of combining \(-4x - 3x\) to produce \(-7x\) and adding constants \(8 + 5\) to give \(13\), resulting in a cleaner and more manageable expression: \(-7x = 13\). Simplifying ensures the expression is in a state where manipulating or solving it is straightforward and free of unnecessary complexity.