Linear Equation
Solve simple linear equations of the form ax + b = 0 instantly. Includes a graph visualization of the line and the solution.
Formula
x = -b / a
To solve ax + b = 0, subtract b from both sides (ax = -b) and then divide by a.
Worked Examples
Example 1: Simple Equation
Problem:Solve 2x - 4 = 0.
Solution:Add 4 to both sides to isolate the x term: 2x = 4 Divide both sides by the coefficient 2: x = 4/2 = 2 Check by substituting back: 2(2) - 4 = 4 - 4 = 0. Correct. Graphically, y = 2x - 4 crosses the x-axis at x = 2.
Result:x = 2
Example 2: Negative Coefficient
Problem:Solve -3x + 9 = 0.
Solution:Subtract 9 from both sides: -3x = -9 Divide by -3. Dividing two negatives gives a positive: x = -9 / -3 = 3 Check: -3(3) + 9 = -9 + 9 = 0. Correct. Note that dividing by a negative flips an inequality sign, but it never flips an equation - both sides stay equal.
Result:x = 3
Example 3: Fractional Result
Problem:Solve 4x + 5 = 0.
Solution:Subtract 5 from both sides: 4x = -5 Divide by 4: x = -5/4 = -1.25 Check: 4(-1.25) + 5 = -5 + 5 = 0. Correct. The exact form -5/4 is preferable to the decimal when the answer feeds into further algebra, since not every fraction terminates.
Result:x = -1.25 (exactly -5/4)
Frequently Asked Questions
What is a linear equation?
A linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
How do you solve ax + b = 0?
To find x, you need to isolate it. First subtract b from both sides (ax = -b), then divide by a (x = -b/a).
What does the solution represent?
The solution x is the x-intercept (root) of the linear function y = ax + b. It represents the point where the line crosses the horizontal x-axis.
What if "a" is zero?
If a=0, the equation becomes b=0. If b is also 0, any x works (infinite solutions). If b is not 0, it is a contradiction (no solution).
What is slope-intercept form?
y = mx + b is slope-intercept form. In our calculator (ax+b=0), "a" acts like the slope of the line y=ax+b.
How are linear equations used in real life?
They are used for rate problems (speed/distance), budgeting (fixed cost + variable cost), and converting units (like Celsius to Fahrenheit).
How do I check my answer?
Substitute your value for x back into the original equation. If the left side equals the right side (0), your answer is correct.
Can a linear equation have two solutions?
No, a linear equation in one variable can have exactly one solution, no solution, or infinite solutions. It cannot have exactly two.
What is the standard form?
For two variables, standard form is Ax + By = C. For one variable, it is ax + b = 0.
Background & Theory
Linear Equation in One Variable:
The standard form is ax + b = 0.
- a: Coefficient (cannot be 0 for a unique solution)
- x: Variable
- b: Constant
Solving Steps:
- Isolate the variable term: Move the constant 'b' to the other side by subtracting it. $$ax = -b$$
- Solve for x: Divide both sides by the coefficient 'a'. $$x = \\frac{-b}{a}$$
Geometric Interpretation:
If you graph the function $f(x) = ax + b$, the solution represents the x-intercept (the zero) of the function.
Types of Solutions:
- Conditional: One solution (a ≠ 0).
- Identity: Infinite solutions (a = 0, b = 0).
- Contradiction: No solution (a = 0, b ≠ 0).