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Understanding algebraic manipulation is an essential skill for solving linear equations. It involves using the properties of arithmetic operations to rearrange equations, making them simpler to solve. Let's consider the given exercise involving the equation 7 - 2x = 9.

To isolate the variable x, we need to perform algebraic manipulations. This may include adding or subtracting the same number from both sides of an equation, or multiplying or dividing both sides by the same non-zero number. In this example, 2x is subtracted from 7 on the left side, so to undo this operation and isolate x, we add 2x to both sides, which results in 7 = 2x + 9. Next, we subtract 9 from both sides to get -2 = 2x. Finally, by dividing both sides by 2, we find that x equals -1.

Practice with algebraic manipulation can significantly improve your confidence in solving equations, making it a vital area to focus on in algebra.

The heart of algebra revolves around the ability to solve equations. Equation solving is a process of finding the value of unknown variables that satisfy the given equation. To master this, one must understand the properties of equality and the rules of operation.

Look at the equation we started with: 7 - 2x = 9. After the algebraic manipulations, we arrived at the simplified form 2x = -2. To solve for x, we must isolate the variable on one side. We do this by dividing both sides of the equation by 2, which is the coefficient of x. This yields x = -1, which is the solution to the equation. Remember, balancing the equation by performing the same operation on both sides is crucial to maintaining equality throughout the process of solving.

Verification is the final step in the process of solving an equation. After finding a solution, it's important to check that it actually works in the original equation. This step ensures that our manipulations and calculations are correct.

Using the solution x = -1 from our previous work on the equation 7 - 2x = 9, we substitute -1 for x back into the original equation to verify. This gives us 7 - 2(-1) = 9, which simplifies to 7 + 2 = 9. Since both sides equal 9, the solution is verified. This step is fundamental and should not be omitted. It completes the equation-solving process and assures you that the solution you found is the correct one.