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Set-builder notation is a mathematical way to describe a set by stating the properties that its members must satisfy. It's very useful when talking about solutions to equations, especially systems of equations. For example, if a system of equations has an infinite number of solutions, you can express all these solutions compactly using set-builder notation.
For instance, let's say we have two variables, x and y, meeting the condition y = 2x + 3. Instead of listing all the infinite pairs \((x, y)\) that satisfy the equation, set-builder notation lets us write it like this:
\(\text{{S}} = \{ (x, y) \; | \; y = 2x + 3 \}\)
This reads as

The substitution method is a common technique for solving systems of linear equations. Here's how it works:
1. **Isolate a variable in one of the equations**. This means rearranging the equation to solve for one variable in terms of the other variable(s).
2. **Substitute the expression**. Take the isolated variable's expression and plug it into the other equation. This step reduces the number of variables, making it easier to solve.
3. **Solve the resulting equation**. Solve for the remaining variable.
4. **Substitute back**. Once you have one variable, substitute it back into one of the original equations to find the other variable.
In the given exercise, we started with the system of equations:
\(a - 2b = 16\)
\(b + 3 = 3a\)
We isolated \(b\) in the second equation and substituted this value into the first equation. This reduced our system to a single equation with one variable, which we solved step by step.

A unique solution in a system of linear equations means that there is exactly one pair of values for the variables that satisfy all equations in the system. When you find a unique solution, it means the system is consistent and independent.
In the given exercise, after using the substitution method, we found\( a = -2 \) and \( b = -9 \). We then substituted these values back into both original equations to verify that they indeed satisfy both equations:
1. For \( a - 2b = 16 \):
\(-2 - 2(-9) = 16 \)
This simplifies to:
\( -2 + 18 = 16 \)
Since 16 equals 16, the solution satisfies the first equation.
2. For \( b + 3 = 3a \):
\( -9 + 3 = 3(-2) \)
This simplifies to:
\( -6 = -6 \)
Since -6 equals -6, the solution satisfies the second equation.
Since both equations hold true for \( a = -2 \) and \( b = -9 \), our solution is indeed unique. This concludes that the system is consistent with exactly one solution.