How to Solve a Linear Function Step by Step?

Solving a linear function typically involves finding the value of the independent variable, usually represented by x, when the dependent variable, f(x) or y, is known. This process transforms the function into a solvable linear equation. By following a structured approach, you can efficiently arrive at the correct solution.

Understanding Linear Functions

A linear function is a mathematical relationship that, when graphed, forms a straight line. It is generally expressed in the form f(x) = mx + b or y = mx + b, where:

m is the slope of the line, representing the rate of change.
b is the y-intercept, the point where the line crosses the y-axis.
x is the independent variable.
f(x) (or y) is the dependent variable, representing the output of the function for a given x.

The goal of "solving" a linear function often means determining the specific x-value that corresponds to a given f(x) value, or finding the x-intercept (where f(x) = 0).

The Step-by-Step Process to Solve a Linear Function

Here’s a breakdown of the key steps to solve a linear function, ensuring you isolate the unknown variable:

Step	Action	Description
1	Substitute the Given Value	Replace f(x) (or y) in the function equation with the specific numerical value provided in the problem. This converts the function into a straightforward linear equation.
2	Isolate the Variable (Initial Steps)	Begin isolating the variable (usually x) by performing inverse operations to move constant terms to the opposite side of the equation. This typically involves using addition or subtraction.
3	Continue to Isolate the Variable (Final Steps)	Complete the isolation of the variable by performing the final inverse operation. This usually involves multiplication or division to get x by itself with a coefficient of 1.
4	Confirm Your Solution	Verify the accuracy of your solution by substituting the calculated x-value back into the original linear function. If both sides of the equation are equal, your solution is correct.

Let's delve into each step with more detail.

Step 1: Substitute the Given Value

The first action is to take the given value for f(x) (or y) and plug it directly into your linear function equation. This immediately transforms the function into a solvable linear equation with only one unknown variable.

Example:
If you have the function f(x) = 3x - 7 and you need to solve it when f(x) = 14, you would substitute 14 for f(x):
14 = 3x - 7

Step 2: Isolate the Variable (Initial Steps)

Once you have an equation, your primary goal is to get the variable term by itself on one side of the equality sign. This usually starts by moving any constant terms (numbers without a variable) from the side of the variable to the other side. You achieve this using inverse operations:

If a number is being added to the variable term, subtract it from both sides of the equation.
If a number is being subtracted from the variable term, add it to both sides of the equation.

Continuing the Example (14 = 3x - 7):
To move the -7 from the right side, add 7 to both sides of the equation:
14 + 7 = 3x - 7 + 7
21 = 3x

Step 3: Continue to Isolate the Variable (Final Steps)

After dealing with addition and subtraction, the variable term might still have a coefficient (a number multiplying it). To get the variable completely by itself (with a coefficient of 1), you'll use more inverse operations:

If the variable is being multiplied by a number, divide both sides of the equation by that number.
If the variable is being divided by a number, multiply both sides of the equation by that number.

Continuing the Example (21 = 3x):
The variable x is being multiplied by 3. To isolate x, divide both sides of the equation by 3:
21 / 3 = 3x / 3
7 = x

So, x = 7 is the solution.

Step 4: Confirm Your Solution

The final and crucial step is to confirm that your calculated value for x is correct. Do this by plugging your solution back into the original linear function. If the equation holds true (both sides are equal), your answer is accurate.

Confirming the Example (x = 7 in f(x) = 3x - 7 when f(x) = 14):
Substitute x = 7 into f(x) = 3x - 7:
f(7) = 3(7) - 7
f(7) = 21 - 7
f(7) = 14

Since the calculated f(7) is 14, which matches the initial given value for f(x), the solution x = 7 is confirmed as correct.

Practical Example: Putting It All Together

Let's solve the linear function f(x) = -2x + 10 for x when f(x) = 4.

Substitute the given value:
4 = -2x + 10
Isolate the variable (initial steps - addition/subtraction):
Subtract 10 from both sides to move the constant term:
4 - 10 = -2x + 10 - 10
-6 = -2x
Continue to isolate the variable (final steps - multiplication/division):
Divide both sides by -2 to get x by itself:
-6 / -2 = -2x / -2
3 = x
Confirm your solution:
Substitute x = 3 back into the original function f(x) = -2x + 10:
f(3) = -2(3) + 10
f(3) = -6 + 10
f(3) = 4

Since f(3) = 4 matches the given f(x) value, our solution x = 3 is correct.

By following these systematic steps, solving any linear function becomes a clear and manageable process, ensuring accuracy and a thorough understanding of the underlying algebraic principles.