Question

Rewrite the following implicit equations in the explicit form, \(y=f(x)\). $$2 x-y+4=0$$

Short answer

The explicit form of the equation is: \(y = 2x + 4\).

Step-by-step solution

Isolate the variable y

To rewrite the implicit equation in the explicit form, isolate the variable y on one side of the equation. Start by adding y to both sides of the equation to get y on one side.

Solve for y

After isolating y, subtract 2x from both sides and then subtract 4 from both sides to solve for y.

Write the explicit equation

Now that y is isolated and we have solved for it in terms of x, write down the final explicit form of the equation, where y is a function of x.

Key concepts

Implicit to Explicit Equation

Converting an equation from implicit to explicit form is like revealing the clear relationship between the variables involved. In implicit equations, all variables are mingled together, like a mixed fruit basket. Explicit equations, on the other hand, are like neatly arranged fruit trays, where each variable's role is clear and straightforward, typically with the 'y' isolated as the subject.
For example, in the equation, \(2x-y+4=0\), 'y' is mixed with 'x'. Our goal is to rewrite it so 'y' is expressed solely in terms of 'x', which looks like \(y=f(x)\). This process benefits students by providing a clearer, more direct understanding of how 'y' depends on 'x'. This is crucial for graphing functions, predicting outcomes, and applying calculus operations later on.

Isolating Variables

Isolating variables is a fundamental skill in algebra that's akin to finding Waldo in a 'Where's Waldo?' book. It's about moving everything apart from the variable of interest (often 'y') to the other side of the equation.
To isolate 'y' in our equation, we performed a dance of shuffling terms from one side to the other using operations that keep the equation balanced. The critical rule here is whatever you do to one side of the equation, you do to the other. We started by adding 'y' to both sides, gently nudging it away from the company of the other terms.

Solving for y

Once we've given 'y' its own space, solving for it becomes simpler. This is like knowing the destination and figuring out the most efficient route to get there.
The equation now feels incomplete without 'y' clearly defined. So, we subtract '2x' and then '4' from both sides to solve for 'y'. Each step is calculated to bring 'y' closer to being by itself on one side of the equal sign, revealing the relationship between 'y' and 'x' in its simplest form.

Algebraic Manipulation

Algebraic manipulation is the art of transforming equations into more useful forms, using a set of mathematical moves. It's like a game of chess with numbers and variables, each move strategic and governed by rules to maintain equality.
In our case, the manipulation involved performing additive inverses (adding the same number to each side to cancel out a term) and getting 'y' all alone on the stage. The final result of this performance is an explicit equation where 'y' shines as the solo artist, defined solely in terms of 'x'. These maneuvers require practice, but once mastered, they empower students to navigate through the maze of algebra with confidence.