Problem 22 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$\frac{1}{4} y+3=-5 x$$

Step-by-Step Solution

Answer

Rewritten equation with \(y\) as a function of \(x\) is \(y = -20x -12\)

Step 1: Move \(x\) term to the right side

To start, the goal is to leave \(y\) on one side of the equation. To do this, take \(5x\) on both sides of the equation to eliminate it from the right side. The equation becomes \(\frac{1}{4}y + 5x + 3 = 0\).

Step 2: Isolate \(y\)

Next, subtract 3 from both sides to isolate the \(y\) term, giving the equation \(\frac{1}{4}y + 5x = - 3\).

Step 3: Make \(y\) the subject

Finally, to make \(y\) the subject, multiply every term by 4 to clear the fraction in front of \(y\). The final equation becomes \(y + 20x = -12\), which when rearranged will give \(y = -20x -12\).

Key Concepts

Solving Equations

Solving equations is like solving a mystery. You want to find the value of a variable, which is like finding the missing piece of a puzzle. The original problem involves solving an equation where we need to express the variable \( y \) in terms of \( x \). This process changes the form of the equation but keeps the values of \( y \) and \( x \) true to the initial condition.

To begin solving: - Rearrange the equation until you isolate the desired variable on one side. - Perform the same operation on both sides to maintain balance, just like a balanced scale.

For example, if you add or subtract a term from one side, do the same to the other. This is crucial as it ensures the equation remains valid. Keep practicing, and you'll get better at using these basic steps to solve more complex equations.

Functions

Understanding functions is a key concept in mathematics, especially in algebra. A function is like a special rule that assigns each input value exactly one output value. In this exercise, we are transforming an equation so that \( y \) becomes a function of \( x \), meaning \( y \) is expressed in terms of \( x \).

This basically involves displaying the relationship between \( y \) and \( x \) clearly. After solving, if you plug in different numerical values for \( x \), you will get a specific value for \( y \). - This also makes it easy to graph these relationships, as each \( x \) produces its own corresponding \( y \). - The rewritten form, \( y = -20x - 12 \), indicates \( y \) is a linear function of \( x \). Letting us spot a straight line if we plotted these values on a graph.

The function approach helps you see how changing one variable influences the other.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying expressions to solve equations. It’s much like reshuffling a deck of cards while ensuring you still play by the rules. With these maneuvers, you can reveal the variable you’re solving for.

In this particular problem, a few manipulative steps were used:

• Moving terms: Terms like \( 5x \) were shifted from one side to another to help isolate \( y \).
• Clearing fractions: Multiplying every term by 4 wiped out the fraction \( \frac{1}{4} \) to make calculations easier.

These are common techniques when dealing with algebraic expressions. Applying them not only simplifies solving equations but also ensures your final expression is neat and clear. The goal is always to make one variable the clear subject or make the expression as simplified as possible.

With these tools, algebra can become a handy language for solving various mathematical problems and real-world puzzles.