91Ó°ÊÓ

In algebra, isolating the variable means getting the variable you are solving for by itself on one side of the equation. Here, we want to solve the equation for \( y \). The given equation is:

\( 4x - 3y = 6 \)

First, we need to isolate the term with \( y \). To do this, we move all other terms to the opposite side of the equation.

To get rid of the \( 4x \) on the left side, subtract it from both sides:

\( 4x - 3y - 4x = 6 - 4x \)

This simplifies to:

\( -3y = 6 - 4x \)

Now we have isolated the term with \( y \), making it easier to solve.

Remember, you can isolate a variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Simplifying fractions makes them easier to understand and work with. Once you isolate the \( y \) term, you usually need to simplify the equation by getting \( y \) completely by itself.

From our isolated equation:

\( -3y = 6 - 4x \)

we divide everything by \( -3 \):

\[ y = \frac{6 - 4x}{-3} \]

This fraction can be simplified. Notice the negative sign in the denominator. We can distribute that negative sign:

\[ y = \frac{6}{-3} + \frac{-4x}{-3} \]

Simplifying each term inside the fraction gives us:

\[ y = -2 + \frac{4x}{3} \]

Or combining into a single fraction:

\[ y = \frac{4x - 6}{3} \]

Simplifying fractions correctly ensures the solution is in its simplest form, making it easier to understand and check.

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In solving for \( y \) in the equation \( 4x - 3y = 6 \), we use several algebraic techniques:

First, we isolate the \( y \) term by subtracting \( 4x \) from both sides of the equation:

\( 4x - 3y - 4x = 6 - 4x \)

Simplifies to:

\( -3y = 6 - 4x \)

Next, we solve for \( y \) by dividing every term by \( -3 \):

\[ y = \frac{6 - 4x}{-3} \]

Finally, we simplify the fraction by distributing the negative sign:

\[ y = \frac{-6 + 4x}{3} \]

Or equivalently,

\[ y = \frac{4x - 6}{3} \]

Each step follows logical rules, making it easier to understand and arrive at the correct solution. Mastering these manipulations is key to solving algebra problems efficiently.