Problem 14 - Algebra 1

Question

Write the equation in the form \(a x+b=0\). Then write the related function \(y=a x+b\). $$7 x-3=3$$

Step-by-Step Solution

Answer

The equation in the form \(ax+b=0\) is \(7x-6=0\). The function \(y=ax+b\) related to this equation is \(y=7x-6\).

Step 1: Rewrite the Equation in the Form \(ax+b=0\)

Initially, we are given the equation \(7x-3=3\), which isn't in the form \(ax+b=0\). To alter the equation to that format, both sides must balance. Therefore, we add 3 to both sides to keep the equation balanced. This gives us \(7x-3+3=3+3\), which simplifies down to \(7x=6\). Subtract 6 from both sides results in \(7x-6=0\). Thus, \(a=7\) and \(b=-6\).

Step 2: Write the Related Function \(y=ax+b\)

The second part of the exercise is to change the newly formulated equation into function form \(y=ax+b\). Here we substitute \(a\) and \(b\) with the numbers derived in Step 1. Hence, \(y=ax+b\) becomes \(y=7x-6\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is essentially the 'toolkit' for working with equations. It involves rearranging, simplifying, and solving equations using a variety of techniques. Critical operations include adding, subtracting, multiplying, and dividing terms on both sides of an equation to maintain balance.

For instance, in the exercise given, transforming the equation into the form \( ax+b=0 \) requires manipulating the original equation \( 7x-3=3 \) so that one side equals zero. This involves the simple yet foundational concept of 'doing to one side what you do to the other' to keep the equation's balance. Adding 3 to both sides neutralizes the '-3' on the left side and changes the equation to \( 7x=6 \)—a clear example of algebraic manipulation in action.

Once here, subtracting 6 from both sides gives us the standard form, \( 7x-6=0 \), showcasing further manipulation by isolating terms and setting the stage for the introduction of function notation. It is essential to comprehend this toolkit to be adept at attacking all forms of algebraic problems.

Linear Equations

Linear equations are one of the most fundamental concepts in algebra. A linear equation is an equation for a straight line and is typically written in the form \( y=mx+c \), where \( m \) represents the slope, and \( c \) is the y-intercept. However, another form of linear equation is the standard form, \( ax+b=0 \), which directly tells us that the sum of the terms, when multiplied by their respective coefficients, equals zero.

In the given problem, when we converted \( 7x=6 \) to \( 7x-6=0 \), we achieved a standard form linear equation. This form is especially useful for certain algebraic operations and is also a preferred way to represent linear equations for some mathematicians due to its simplicity and structure.

Becoming skilled in working with linear equations in various forms is crucial, as these are not just abstract concepts but represent real-world situations where relationships between quantities are linear, like converting currencies or calculating the slope of a roof.

Function Notation

Function notation offers a way to represent functions algebraically. When you see \( y=f(x) \) or \( y=ax+b \), 'y' is essentially the output of the function given the input 'x'. The notation emphasizes the dependent relationship between variables, with the value of 'y' being determined by the assigned value of 'x'.

The exercise involves converting an equation into function notation, which serves as a bridge between purely algebraic expressions and functional relationships. From the standard form \( 7x-6=0 \), rewriting this as \( y=7x-6 \) immediately indicates that for every input \( x \) there is an output \( y \) defined by the operation \( 7x-6 \).

Function notation is not just a notational convenience but a significant conceptual leap in mathematics. It establishes a clear understanding of functions as processes that take inputs and produce outputs—a core concept in not just algebra, but in the broader landscape of mathematical analysis and beyond.