AI Linear Equation Solver

The balance method in four moves

Think of an equation as a set of scales. Anything you add to or remove from one pan you must add to or remove from the other. The solver applies four moves:

1. Simplify each side by expanding brackets and collecting like terms.
2. Gather the x-terms on one side and the numbers on the other.
3. Divide by the coefficient of x to leave x on its own.
4. Check the answer in the original equation.

One solution, no solution or infinite solutions

After simplifying, a linear equation falls into one of three cases. Most have exactly one answer. If the x-terms cancel and you are left with a false statement such as 6 = 5, there is no solution, because the two sides are parallel lines that never meet. If you are left with a true statement such as −3 = −3, there are infinitely many solutions, because both sides describe the same line. The solver reports which case you have and explains why.

Where linear equations appear

Linear equations model anything that grows at a constant rate: taxi fares, phone plans, simple interest and unit conversions. Turning a story into an equation is covered in the word problem solver. When two linear equations share the same unknowns, use the system of equations solver. If the highest power is 2, switch to the quadratic equation solver.

Worked examples

Example 1: Solve 5x + 7 = 2x − 8

1. Subtract 2x from both sides: 3x + 7 = −8.
2. Subtract 7 from both sides: 3x = −15.
3. Divide by 3: x = −5.
4. Check: 5(−5) + 7 = −18 and 2(−5) − 8 = −18 ✓

Final answer: x = −5

Example 2: Solve 2(x + 3) = 2x + 5

1. Expand the bracket: 2x + 6 = 2x + 5.
2. Subtract 2x from both sides: 6 = 5.
3. This is false whatever x is.

Final answer: No solution

Example 3: Solve 3(x − 1) = 3x − 3

1. Expand the bracket: 3x − 3 = 3x − 3.
2. Both sides are identical, so the statement is always true.

Final answer: Infinitely many solutions

Example 4: Solve x/2 + x/3 = 10

1. Multiply every term by the common denominator 6: 3x + 2x = 60.
2. Combine: 5x = 60.
3. Divide by 5: x = 12.
4. Check: 6 + 4 = 10 ✓

Final answer: x = 12

Common mistakes to avoid

• Moving a term across the equals sign without changing its sign.
• Dividing only one term by the coefficient instead of the whole side.
• Forgetting to multiply the right-hand side when clearing fractions.
• Stopping at 3x = −15 and giving that as the answer.
• Assuming every equation has exactly one solution.

Frequently asked questions

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