Question

Solve the equation. $$\frac{y}{-2}=\frac{5}{4}$$

Short answer

The solution to the equation is \(y = -\frac{5}{2}\).

Step-by-step solution

Isolate Variable

To isolate the variable \(y\) on one side, multiply both sides of the equation by \(-2\). Thus, the equation now becomes: \(-2*\frac{y}{-2} = -2*\frac{5}{4}\).

Simplify Equation

After multiplication, the equation simplifies to: \( y = -\frac{5}{2}\).

Key concepts

Isolating Variables

Understanding how to isolate variables is an essential skill when it comes to solving linear equations. Imagine it as finding a way to have the unknown, which is often represented by a letter such as 'x', 'y', or 'z', stand alone on one side of the equation. This is achieved through performing operations that 'reverse' what is currently being done to the variable.

For instance, in our exercise, we are given a fractional equation \(\frac{y}{-2}=\frac{5}{4}\). To isolate the variable \(y\), we need to eliminate the fraction. This can be done by multiplying both sides of the equation by the denominator of the variable's fraction, which is -2 in this case. By doing so, the aim is to ultimately have the variable on one side and a simplified answer on the other side. This process is crucial as it forms the basis for solving more complex equations where multiple steps might be required to isolate the variable.

Equation Simplification

Simplifying an equation means to perform steps that make the equation less complicated and more manageable. It usually involves combining like terms, reducing fractions, or cancelling out factors. Simplification is not just about making the equation look neater; it also makes the next steps in solving the equation more transparent and less error-prone.

In our given problem, after isolating \(y\), it's essential to simplify the equation to find the variable's value. Multiplying both sides by -2, as mentioned, we get rid of the fraction, and the equation becomes \(y = -\frac{5}{2}\). Here, multiplication by -2 and simplification are intertwined; the multiplication itself simplifies the equation by clearing the fraction. It's a critical process that transforms the equation from one form to a simpler equivalent that is easier to interpret and solve.

Fractional Equations

Fractional equations involve variables that appear in the numerator or denominator of a fraction. They can be intimidating, but the same principles of equation solving apply. The key is to clear the fractions early on, making the rest of the problem similar to any other linear equation. One common method is to multiply through by the Least Common Denominator (LCD) to cancel out the fractions.

In our exercise, the variable \(y\) is in the numerator and the LCD would be the number -2. Multiplying through by -2, we effectively get rid of the fractional component, leading to a simpler, non-fractional form. Understanding how to manipulate fractional equations is important because it's a tool you'll use frequently, especially in algebra and pre-calculus. It's also a good habit to check your solutions by plugging them back into the original fractional equation to ensure they work.