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When faced with a linear equation like \-7 t+11=-73\, the goal is to isolate the variable (in this case, \(t\)) on one side of the equation to find its value. This means we need to perform operations on the equation that will leave \(t\) by itself. The principle behind isolating the variable is the idea of balance - whatever you do to one side of the equation, you must do to the other side. This maintains the equality of the equation.

To isolate \(t\), you'll first remove any added or subtracted terms from the same side as the variable. In our exercise, this means subtracting 11 from both sides, turning \-7t + 11 - 11 = -73 - 11\ into \-7t = -84\. Following this, to undo the multiplication of \(t\) by -7, you divide both sides by -7, resulting in the variable standing alone: \(t = 12\). Understanding how to manipulate an equation to isolate the variable is a fundamental skill in algebra which applies to all types of equations.

Solving linear equations involves a series of well-defined steps to find the value of the unknown variable. The process for our example \-7t + 11 = -73\, can be broken down into clear stages for clarity and ease of understanding.

Step 1: Identify the Variable

Recognize the variable to be solved for. In this equation, our variable is \(t\).

Step 2: Undo Addition or Subtraction

Remove any constants from the side with the variable by performing the opposite operation. Here, we subtract 11 from both sides to eliminate the +11 next to \(t\).

Step 3: Undo Multiplication or Division

If the variable is multiplied or divided by a number, undo this by doing the inverse operation. In our case, we divide by -7 to counteract the multiplication by -7.

Step 4: Simplify and Solve

After isolating the variable, simplify the equation if needed to find the value of the variable. Here, it simplifies to \(t=12\).
Following the steps systematically leads to an accurate and understandable solution to the linear equation.

After finding the solution to an equation, it's crucial to verify it to ensure it's correct. This involves substituting the value of the variable back into the original equation and checking if it satisfies the equation.

For the equation \-7t + 11 = -73\, after solving, we found that \(t = 12\). To check, replace \(t\) with 12 in the original equation: \-7(12) + 11\, simplifies to \-84 + 11\, which equals \-73\, the same as the right side of our original equation. If the left and right sides match after the substitution, the solution is correct. This check acts as a confirmation and is a good habit to ensure accuracy in algebra. It can prevent simple errors that might occur during the process of solving the equation. Always remember, checking your solution is just as important as finding it.