ax+b=c

2025

2 min read 31-03-2025

Meta Description: Learn how to solve the algebraic equation ax + b = c step-by-step. This comprehensive guide covers various scenarios and provides practice problems to solidify your understanding. Master this fundamental equation and build your algebra skills! (158 characters)

Understanding the Equation ax + b = c

The equation ax + b = c is a fundamental algebraic equation. It represents a linear equation with one variable, 'x'. Understanding how to solve for 'x' is crucial for more advanced algebraic concepts. This equation involves three constants: 'a', 'b', and 'c', where 'a' cannot be zero (otherwise, it wouldn't be a linear equation).

What do a, b, and c represent?

a: This is the coefficient of the variable 'x'. It represents the multiplier of 'x'.
b: This is a constant term added to the 'ax' term.
c: This is the constant term on the other side of the equation, representing the total value.

Step-by-Step Solution to ax + b = c

Solving for 'x' involves isolating it on one side of the equation. Here’s a breakdown of the steps:

1. Subtract 'b' from both sides:

The first step is to eliminate 'b' from the left side of the equation. To do this, subtract 'b' from both sides of the equation, maintaining balance:

ax + b - b = c - b

This simplifies to:

ax = c - b

2. Divide both sides by 'a':

Now, 'x' is multiplied by 'a'. To isolate 'x', divide both sides of the equation by 'a':

ax / a = (c - b) / a

This gives us the solution for 'x':

x = (c - b) / a

Important Considerations

a ≠ 0: Remember, 'a' cannot be zero. Division by zero is undefined.
Negative Values: The values of 'a', 'b', and 'c' can be positive, negative, or zero (except for 'a'). Be mindful of the rules for working with negative numbers.
Fractions and Decimals: The solution for 'x' might result in a fraction or decimal. It's important to be comfortable working with these types of numbers.

Example Problems

Let's solidify our understanding with some examples:

Example 1:

Solve for 'x' in the equation 2x + 5 = 11

Subtract 5 from both sides: 2x = 6
Divide both sides by 2: x = 3

Example 2:

Solve for 'x' in the equation -3x - 7 = 8

Add 7 to both sides: -3x = 15
Divide both sides by -3: x = -5

Example 3:

Solve for 'x' in the equation 4x + 12 = 4

Subtract 12 from both sides: 4x = -8
Divide both sides by 4: x = -2

Applications of ax + b = c

This seemingly simple equation has broad applications in various fields:

Physics: Solving for velocity, acceleration, or time in kinematic equations.
Engineering: Calculating unknown variables in design equations.
Finance: Determining interest rates or future values.
Computer Science: Solving for unknown variables in algorithms and calculations.

Mastering this equation provides a solid foundation for more complex algebraic problems. Practice these examples, and you'll quickly become comfortable solving for 'x' in ax + b = c equations!