91Ó°ÊÓ

Linear functions are mathematical expressions where each variable is raised to the power of one, creating a straight line when graphed. These functions have the general form \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.
Understanding the slope is crucial as it describes the steepness and direction of the line:

• A positive slope means the line ascends as you move from left to right.
• A negative slope means the line descends as you move from left to right.
• A zero slope indicates a horizontal line.

In our example, \(f(x) = 4x - 3\), we can see that the slope \(m\) is 4, indicating a relatively steep ascent.
This means, every time \(x\) increases by 1, the function’s value increases by 4. The intercept \(-3\) tells us that when \(x = 0\), the value of \(f(x)\) is \(-3\). Understanding these parameters helps make sense of how changes in \(x\) impact \(f(x)\).

The substitution method is a technique where one substitutes known values into an equation to find unknowns.
In the context of solving equations, it often involves replacing a variable with a specific value given by another condition.
In our original exercise, we were asked to find the value of \(x\) when \(f(x) = -2\).
Using the substitution method, we replace \(f(x)\) in the equation \(4x - 3\) with \(-2\):

• Given: \(f(x) = 4x - 3\)
• Replace \(f(x)\) with \(-2\), so it becomes \(-2 = 4x - 3\)

This step transforms the problem from finding the function’s value to a straightforward algebraic equation we can solve.
Substitution is an essential tool for simplifying problems and making complex equations more manageable. Understanding this approach can enhance problem-solving skills across various math topics.

Isolating a variable involves manipulating an equation until the variable of interest stands alone on one side of the equation.
This process makes it simple to find the value of the variable. It’s a core part of solving equations, especially linear ones.
In our task, we aimed to isolate \(x\) in the equation \(-2 = 4x - 3\). Here's how we did it:

• Add 3 to both sides: \(-2 + 3 = 4x\), simplifying to \(1 = 4x\).
• Divide both sides by 4 to isolate \(x\): \(\frac{1}{4} = x\).
• Thus, \(x = 0.25\).

By following these steps, we isolate \(x\) successfully, finding its value is \(0.25\).
Isolating the variable often requires inverse operations, such as adding, subtracting, multiplying, or dividing. Each step reverses the operations affecting the variable, guiding us to the solution.