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Linear functions are a type of function that create a straight line when graphed on a coordinate plane. They can be represented by the equation in the form of \f(x) = mx + b\, where \(m\) is the slope and \(b\) is the y-intercept.

In the given exercise, our function is \(f(x) = 2.3 - 1.5x\). Here, the slope \(m\) is -1.5 and the y-intercept \(b\) is 2.3. This means for every 1 unit increase in \(x\), the value of \(f(x)\) decreases by 1.5. You can think of a linear function as a way to calculate the output for any input \(x\), by following a very straightforward rule or pattern.

Understanding linear functions is crucial, as they appear in many real-life scenarios, such as calculating expenses over time, predicting trends, and much more.

Isolating a variable means getting the variable on its own on one side of the equation. This is essential in solving equations, regardless of complexity.

To isolate the variable in the equation \(2.3 - 1.5x = 10\), we performed the following steps:

• First, subtract 2.3 from both sides: \(2.3 - 1.5x - 2.3 = 10 - 2.3\)
• This leaves us with \(-1.5x = 7.7\).

By isolating \(x\), we essentially simplified the problem to a single, straightforward operation.

Solving for a variable is the process of finding the value of the variable that makes the equation true. Once we've isolated the variable, the final step is solving for it.

In our example, after isolating the variable, we had \(-1.5x = 7.7\). To solve for \(x\):

• Divide both sides by -1.5: \(x = \frac{7.7}{-1.5}\)
• This simplifies to \(x = -5.13\)

In simple terms, we determine the value of \(x\) that makes the original equation balance.

Solving for variables is a fundamental skill in algebra involving manipulating equations to discover unknown quantities. This is widely applicable in science, engineering, economics, and even daily problem-solving.