91Ó°ÊÓ

Algebraic manipulation is the process of rearranging and simplifying equations or expressions using a variety of techniques. The goal is to isolate the variable you're solving for, making the solution clearer.

For example, in the given exercise \(4y - 2 = -16 - 3y\), algebraic manipulation begins with combining like terms on both sides of the equation. Like terms are terms that have the same variables raised to the same powers. Here, we can move the term containing 'y' to one side by adding \(3y\) to both sides, resulting in \(4y + 3y = -16 + 2\).

Next, the coefficients of 'y' are added together (4 + 3), and the constants on the right side are added as well (-16 + 2), further simplifying the equation. By executing this basic step of algebraic manipulation effectively, we move closer to finding the value of 'y'.

Once you have manipulated an algebraic equation, the next step is to solve it to find the value of the variable. The goal of equation solving is to determine the single value or range of values that satisfy the equation.

Continuing with our exercise, after simplifying the like terms, we are left with \(7y = -14\). From here, the aim is to isolate 'y' by performing the inverse operation to what has been applied to it. Since 'y' is multiplied by 7, we divide both sides of the equation by 7 to solve for 'y'.

\(\frac{7y}{7} = \frac{-14}{7}\)
\(y = -2\)

Solving equations involves understanding inverse operations: addition pairs with subtraction, multiplication pairs with division. Recognizing this relationship is crucial to solving any linear equation efficiently.

After finding a solution to an equation, it is important to check that it is correct. Checking solutions confirms that the answer satisfies the original equation, ensuring there were no mistakes during manipulation or solving.

To check the solution \(y = -2\) from our exercise, we substitute \(-2\) back into the original equation for 'y':
\((4 \cdot (-2)) - 2 = -16 - (3 \cdot (-2))\)
By simplifying both sides, we should get an equality. If each side equals the same number, the solution is correct. For this example, the left-hand side becomes \(-8 - 2\), which simplifies to \(-10\), and the right-hand side becomes \(-16 + 6\), which also simplifies to \(-10\). Since both sides equal \(-10\), this confirms that \(y = -2\) is indeed the correct solution.

Verifying solutions cultivates good mathematical practice and assures students of their mastery over the concepts involved.