Linear Equation Solver (ax + b = c)
Enter the coefficients and constants for your linear equation in the form ax + b = c to find the value of x.
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Understanding Linear Equations and How to Solve Them
Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. One of the most common and essential concepts in algebra is the linear equation. A linear equation is an algebraic equation in which each term has an exponent of one, and when plotted on a graph, it forms a straight line. Our calculator focuses on solving linear equations in the standard form ax + b = c.
What is a Linear Equation?
A linear equation is an equation that can be written in the form ax + b = c, where:
xis the variable (the unknown value we want to find).ais the coefficient ofx. It's a number that multiplies the variable.bis a constant term on the left side of the equation.cis a constant term on the right side of the equation.
The goal when solving a linear equation is to isolate the variable x on one side of the equation, determining its value.
How to Solve ax + b = c
The process of solving a linear equation involves a series of inverse operations to get x by itself. Here are the steps:
- Subtract 'b' from both sides: To move the constant term
bfrom the left side to the right side, you perform the inverse operation of addition, which is subtraction.ax + b - b = c - bax = c - b - Divide by 'a' on both sides: To isolate
x, you perform the inverse operation of multiplication, which is division. You divide both sides by the coefficienta.ax / a = (c - b) / ax = (c - b) / a
This final formula gives you the value of x.
Special Cases: When 'a' is Zero
There are two important special cases to consider when the coefficient a is equal to zero:
- If
a = 0andb = c: The equation becomes0x + b = b, which simplifies tob = b. This statement is always true, regardless of the value ofx. In this scenario, there are infinite solutions. Any real number forxwill satisfy the equation. - If
a = 0andb ≠ c: The equation becomes0x + b = c, which simplifies tob = c. Since we've established thatbis not equal toc, this statement is false. Therefore, there is no solution forxthat can satisfy the equation.
Examples Using the Calculator
Let's look at some practical examples of how to use the linear equation solver:
Example 1: Basic Linear Equation
Solve: 2x + 5 = 15
- Coefficient 'a':
2 - Constant 'b':
5 - Constant 'c':
15
Using the formula: x = (15 - 5) / 2 = 10 / 2 = 5. The calculator will output x = 5.
Example 2: Negative Numbers
Solve: -3x + 7 = -8
- Coefficient 'a':
-3 - Constant 'b':
7 - Constant 'c':
-8
Using the formula: x = (-8 - 7) / -3 = -15 / -3 = 5. The calculator will output x = 5.
Example 3: Infinite Solutions
Solve: 0x + 10 = 10
- Coefficient 'a':
0 - Constant 'b':
10 - Constant 'c':
10
Since a = 0 and b = c, the calculator will correctly identify that there are infinite solutions.
Example 4: No Solution
Solve: 0x + 4 = 9
- Coefficient 'a':
0 - Constant 'b':
4 - Constant 'c':
9
Since a = 0 and b ≠ c, the calculator will correctly identify that there is no solution.
This linear equation solver is a handy tool for students and anyone needing to quickly verify solutions to basic algebraic problems. By understanding the underlying principles, you can confidently tackle more complex equations.