Problem 19 - Algebra 1

Question

Write the equation in the form \(a x+b=0\). Then write the related function \(y=a x+b\). $$6-\frac{4}{7} x=13+\frac{3}{7} x$$

Step-by-Step Solution

Answer

The given equation transforms to the function \(y = x\).

Step 1: Simplify the Equation

Consider the equation \(6 - \frac{4}{7}x = 13 + \frac{3}{7}x\). Simplify the equation by adding \(\frac{4}{7}x\) to both sides and subtract 13 from both sides. This will lead to \(\frac{7}{7}x = -7\).

Step 2: Put in Ax + B form

Simplify \(\frac{7}{7}x = -7\) to \(x = -7\), which can also be expressed in the form \(1x + 0 = 0\), where \(a=1\) and \(b=0\).

Step 3: Write as a Function

Transform the equation \(1x + 0 = 0\) into the function \(y = 1x + 0\), which simplifies to \(y = x\).

Key Concepts

Linear Equations

Linear equations are foundational to algebra and are equations that represent straight lines when graphed. These equations typically have variables raised to the first power and can have one or more variables, but they do not contain variables that multiply each other or variables in the denominator of a fraction. In their simplest form, linear equations look like:
\[ ax + b = 0 \]
Here, \(a\) and \(b\) are constants, where \(a\) is the slope of the line and \(b\) is the y-intercept. The process of writing a linear equation involves rearranging the terms to obtain a standard form that meets the specified condition, which is often \( ax + b = 0 \).

For instance, the exercise provided asks to transform \[ 6 - \frac{4}{7}x = 13 + \frac{3}{7}x \] into standard form. To do this, we need to move all the variable terms to one side of the equation and the constant terms to the other. Finally, we should aim to have positive coefficients for \(x\) if possible, which is standard in presenting linear equations.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their most basic form without changing the original value or equation. It entails combining like terms, reducing fractions, and applying the distributive property when necessary.

Take the initial equation in the exercise: \[ 6 - \frac{4}{7}x = 13 + \frac{3}{7}x \]. The goal is to isolate the \(x\) variable. We can do this by adding or subtracting terms on both sides such that like terms are combined and constants are on the opposite side of the variable. The initial steps include adding \(\frac{4}{7}x\) to both sides to get rid of the negative \(x\) term on the left, and subtracting 13 from both sides to move the constant on the right side over to the left.
\[ 6 - 13 + \frac{4}{7}x + \frac{3}{7}x = \frac{7}{7}x \]
This leads to simplification, where like terms of \(x\) are combined, and simple arithmetic is performed to find the constants. Here, algebraic simplification helps to clarify the situation by making the equation easier to interpret.

Function Notation

Function notation is a way of representing the relationship between variables, typically using the letter \(f\), though any other letter can be used, such as \(g\) or \(y\). In function notation, an equation like \( y = ax + b \) is written as \( f(x) = ax + b \) since the \(y\) value is determined by the \(x\) value - that is, \(y\) is a function of \(x\). This notation makes it clearer that \(y\) depends on the input \(x\) and allows for easy substitutions and evaluations.

Reflecting on the final step from the solution provided, we rewrite the simplified equation \(1x + 0 = 0\) as a function: \(y = 1x + 0\) which simplifies to \(y = x\). In formal function notation, this could also be expressed as \(f(x) = x\), making it distinct that for every input \(x\), the output \(y\) (or \(f(x)\)) is equal to \(x\). Function notation is useful for evaluating functions at specific points, composing functions, and understanding their behavior.