Algebra Calculator
Solve linear equations (ax + b = c) step-by-step for variable x, isolate terms, and verify solutions by substitution.
Linear Equation Solver (ax + b = c)
Enter coefficients a, b, and constant c.
Algebra Solution Output
Quick Summary
Our algebra calculator solve linear equation steps tool solves linear equations of the form $ax + b = c$ for variable $x$. It isolates the variable using inverse algebraic operations and verifies the solution by substitution.
How It Works: Solving Linear Algebraic Equations
A linear equation represents a straight line relationship where the variable $x$ has an exponent of 1:
1. **Identify Coefficients:** Identify multiplier coefficient $a$, left constant $b$, and right constant $c$.
2. **Isolate Variable Term ($ax$):** Subtract $b$ from both sides: $ax = c - b$.
3. **Divide by Coefficient ($a$):** Divide both sides by $a \implies x = \c - b / a$.
4. **Substitution Verification:** Plug calculated $x$ back into original $ax + b = c$ to verify equality.
Formula Explanation
Your linear algebraic equation solutions follow standard variable isolation rules:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for solving 3x - 5 = 10:
- Step 1 (Write Original Equation): $3x - 5 = 10$.
- Step 2 (Add 5 to Both Sides): $3x - 5 + 5 = 10 + 5 \implies \mathbf{3x = 15}$.
- Step 3 (Divide Both Sides by 3): $\3x / 3 = \15 / 3 \implies \mathbf{x = 5}$.
- Step 4 (Substitute Solution for x): $3(5) - 5 = 15 - 5 = \mathbf{10}$.
- Step 5 (Confirm Equality): $10 = 10$ (Equality holds), delivering **x = 5**!
Calculation Examples: Real-World Scenario Comparison
Compare linear algebraic equations, intermediate steps, and variable solutions:
| Linear Equation | Coefficient (a) | Constant (b) | Constant (c) | Step 1 (ax = c - b) | Solution (x) |
|---|---|---|---|---|---|
| 3x - 5 = 10 | 3 | -5 | 10 | 3x = 15 | x = 5.00 |
| 2x + 4 = 14 | 2 | 4 | 14 | 2x = 10 | x = 5.00 |
| 5x = 25 | 5 | 0 | 25 | 5x = 25 | x = 5.00 |
| -4x + 8 = 0 | -4 | 8 | 0 | -4x = -8 | x = 2.00 |
Benefits of Using the Algebra Calculator
Utilizing this calculator provides essential algebra homework, physics motion, and business break-even advantages:
- Step-by-Step Problem Solving: Shows intermediate isolation steps ($ax = c - b$) so students learn algebraic concepts.
- Substitution Verification Engine: Plugs solution back into equation to ensure 100% mathematical accuracy.
- Physics Motion Equations ($v = u + at$): Solves for acceleration $a$ or time $t$ in kinematics.
- Business Break-Even Analysis ($R = C$): Solves for unit quantity $x$ required to balance revenue and costs.
Frequently Asked Questions (FAQ)
What is a linear algebraic equation?
A linear algebraic equation is an equation of degree 1 (ax + b = c) where the variable x has no power higher than 1.
How do you solve for x in ax + b = c?
Subtract b from both sides (ax = c - b), then divide by a: x = (c - b) / a.
What happens if coefficient 'a' is zero?
If a = 0, the equation reduces to b = c. If b = c, there are infinitely many solutions; if b ≠ c, there is no solution.
How do you check your algebraic solution?
Substitute your calculated value for x back into the original equation. If the left side equals the right side, your solution is correct.
How do you solve 3x - 5 = 10?
Add 5 to both sides: 3x = 15. Divide by 3: x = 5.
How do you solve 2x + 4 = 14?
Subtract 4 from both sides: 2x = 10. Divide by 2: x = 5.
What is an inverse operation in algebra?
Inverse operations undo each other: addition undoes subtraction, and multiplication undoes division.
What is the Golden Rule of algebra?
Whatever operation you perform on one side of an equation, you MUST perform on the other side to maintain equality.
Can x be negative?
YES! Variable x can be positive, negative, zero, or a fraction depending on the equation constants.
How do you handle negative coefficients like -4x + 8 = 0?
Subtract 8: -4x = -8. Divide by -4: x = -8 / -4 = +2.