91Ó°ÊÓ

When solving equations, your objective is to figure out the value of the unknown variable that makes the equation true. In our example, the equation is \(-2y - 10 = 5y + 18\). This is a linear equation because the variable \(y\) is to the power of one, which means the graph of this equation would be a straight line. The process of solving equations usually involves operations that simplify the equation to find the value of the variable. This process often includes variable isolation and algebraic manipulation.

By systematically changing the equation through various operations, you maintain its equality while making it easier to solve. The goal is to have the variable on one side of the equation and numbers on the other, simplifying the pathway to find the numerical value of the variable.

Remember, every operation you apply to the equation should be done to both sides. This balances the equation and preserves equality, a crucial aspect of solving equations.

Variable isolation is a strategy in algebra to get the variable by itself on one side of the equation. The aim is to solve for the variable, in this case, \(y\). In the given problem, we apply several steps to isolate \(y\):

• **Step 1**: Add \(2y\) to both sides to eliminate \(-2y\) on the left, resulting in \(-10 = 7y + 18\).
• **Step 2**: Subtract \(18\) from both sides to move the constant to the opposite side, leading to \(-28 = 7y\).
• **Step 3**: Divide both sides by \(7\) to solve \(y\), which results in \(y = -4\).

Proper variable isolation helps clear out other numbers or terms that might be lumped in with your variable. It's like peeling back layers to get to the core solution.

Algebraic manipulation involves the use of mathematical operations to rearrange equations, so they are in simpler, more handy forms. This is done by employing operations such as addition, subtraction, multiplication, and division.

When solving the equation \(-2y - 10 = 5y + 18\), several algebraic manipulation steps were used:

• Adding and subtracting terms to both sides to cancel out unwanted terms. For example, adding \(2y\) to both sides helped cancel \(-2y\)
• Subtracting \(18\) from both sides helped shift the equation to have \(y\) terms on one side and constants on the other, resulting in \(-28 = 7y\).
• Dividing both sides by \(7\) simplified the equation to solve for \(y\), providing a clear and simple conclusion that \(y = -4\).

Practicing multiple algebraic manipulations helps solve complex equations and makes way for clearer solutions.