Linear Equation Solver
Solve any linear equation for x. Enter the equation in any form and get the answer with full step-by-step working shown.
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What's the key thing to understand about The Golden Rule?
Whatever you do to one side of an equation, do the same to the other side. This maintains equality. To solve 3x + 5 = 14: subtract 5 from both sides → 3x = 9. Divide both sides by 3 → x = 3. Always isolate x by working backwards through the order of operations: undo addition/subtraction first, then undo multiplication/division.
What do I need to know about Equations with Variables on Both Sides?
For 5x + 3 = 2x + 12: first collect all x terms on one side. Subtract 2x from both sides: 3x + 3 = 12. Then subtract 3: 3x = 9. Divide by 3: x = 3. The strategy: move smaller x coefficient to the other side to keep x coefficient positive, making the arithmetic easier to follow.
What should I know about Equations with Fractions?
Multiply every term by the LCM of all denominators to eliminate fractions first. For x/3 + 2 = 5: multiply everything by 3 → x + 6 = 15 → x = 9. For x/2 + x/3 = 5: LCM is 6, multiply by 6 → 3x + 2x = 30 → 5x = 30 → x = 6. Eliminating fractions at the start avoids errors with fraction arithmetic during solving.
What's the key thing to understand about Always Verify Your Answer?
After solving, substitute your answer back into the original equation to check it satisfies both sides. For 3x + 5 = 14 with x = 3: left side = 3(3) + 5 = 9 + 5 = 14 = right side ✓. Verification catches arithmetic errors and builds confidence. In exams, showing verification often earns an additional mark even if the answer is wrong.
What's the difference between No Solution and Infinite Solutions?
Not every linear equation has exactly one answer. When you simplify an equation and the variable terms cancel out completely, you're left with a statement about constants alone, and there are two possible outcomes. If the remaining statement is false — such as 5 = 8 — the equation has no solution at all; no value of x can make the original equation true, which happens when both sides describe lines with the same slope but different intercepts (parallel lines that never meet). If the remaining statement is true — such as 5 = 5 — the equation has infinitely many solutions; every real number satisfies it, which happens when both sides are really the same equation in disguise (identical lines). For example, 2x + 3 = 2x + 7 simplifies to 3 = 7 (false, no solution), while 2x + 3 = 2x + 3 simplifies to 3 = 3 (true, infinite solutions). Recognising these two special cases prevents the frustrating experience of hunting for an x value that was never going to exist, or incorrectly treating an identity as if it had a single answer.
What do I need to know about Turning Word Problems into Linear Equations?
Most real-world uses of linear equations start as a sentence, not an equation. The key skill is translating words into algebraic structure: 'is' or 'equals' becomes =, 'more than' or 'increased by' becomes +, 'less than' becomes −, and 'times' or 'of' becomes ×. For example, 'three times a number, plus 7, equals 22' becomes 3x + 7 = 22, which solves to x = 5. A classic mixture problem — 'a plumber charges a $45 call-out fee plus $60 per hour, and the total bill was $225' — becomes 45 + 60h = 225, giving h = 3 hours. The strategy is to assign a variable to the unknown quantity first, identify the fixed amount (constant) and the amount that changes per unit (coefficient), and write the equation reflecting the total. Once set up correctly, solving follows the same golden rule of balancing both sides — the translation step, not the algebra, is usually where word problems go wrong.
What do I need to know about Linear Equations with Decimals?
Linear equations containing decimals can be solved with the same balancing steps used for whole numbers and fractions, but it often helps to first multiply every term by a power of 10 to clear the decimals entirely. For 0.25x + 1.5 = 3.75, multiplying every term by 100 converts it to 25x + 150 = 375, which is easier to work with: subtract 150 to get 25x = 225, then divide by 25 to get x = 9. Choose the smallest power of 10 that clears every decimal in the equation (based on whichever term has the most decimal places) to avoid working with unnecessarily large numbers. This technique is especially useful for equations drawn from real contexts like currency, measurements, or percentages, where decimal coefficients are the norm rather than the exception.
Common Linear Equation Forms Reference Table
Linear equations show up in several standard forms depending on what information you're given. Recognizing the form tells you the fastest way to extract the slope and solve:
| Form | General Equation | When You'd Use It |
|---|---|---|
| Slope-Intercept | y = mx + b | You know the slope and y-intercept directly |
| Point-Slope | y − y₁ = m(x − x₁) | You know the slope and one point |
| Standard Form | Ax + By = C | Useful for finding intercepts quickly |
| Two-Point Form | (y−y₁)/(x−x₁) = (y₂−y₁)/(x₂−x₁) | You know two points on the line, not the slope |
Every one of these describes the same family of straight-line relationships; they're just algebraically rearranged for convenience. A linear equation solver typically works with the "one variable, one equation" case (like 3x + 5 = 17), which is a special, simpler instance of the same balancing principle used to rearrange any of the forms above.
Worked Example: A Multi-Step Equation with Fractions and Variables on Both Sides
Solve: (2x − 3)/4 + 5 = (x + 1)/2 − 2x. Step 1 — clear denominators by multiplying every term by the LCD, which is 4: (2x − 3) + 20 = 2(x + 1) − 8x. Step 2 — expand: 2x − 3 + 20 = 2x + 2 − 8x, which simplifies to 2x + 17 = −6x + 2. Step 3 — collect x terms on one side: 2x + 6x = 2 − 17 → 8x = −15. Step 4 — divide: x = −15/8 = −1.875. Check by substituting x = −1.875 back into both sides: LHS = (2(−1.875) − 3)/4 + 5 = (−3.75 − 3)/4 + 5 = −6.75/4 + 5 = −1.6875 + 5 = 3.3125. RHS = (−1.875 + 1)/2 − 2(−1.875) = −0.875/2 + 3.75 = −0.4375 + 3.75 = 3.3125. Both sides equal 3.3125, confirming the solution is correct. This "clear fractions first, then collect variables" strategy works for any linear equation no matter how many fraction and variable terms it has, as long as you apply the same operation to every single term on both sides.
Solving a System of Two Linear Equations by Substitution
Beyond solving a single equation for one variable, linear equations often appear in pairs that must be solved together to find where two lines intersect. Solve the system: y = 2x + 3 and y = −x + 9. Since both are already solved for y, set them equal to each other: 2x + 3 = −x + 9. Add x to both sides: 3x + 3 = 9. Subtract 3: 3x = 6. Divide by 3: x = 2. Substitute x = 2 back into either original equation: y = 2(2) + 3 = 4 + 3 = 7. So the two lines intersect at the point (2, 7). Check against the second equation: y = −(2) + 9 = 7 ✓ — both equations agree. Geometrically, this point is the single location where the two straight lines cross; if the two equations had the same slope but different intercepts, there would be no solution (parallel lines that never meet), and if they had the same slope and the same intercept, every point on the line would be a solution (the two equations describe the identical line). This substitution technique generalizes directly to systems written in standard form (Ax + By = C) by first isolating y in one equation, then substituting that expression into the other — the exact method used by graphing calculators internally before they plot the intersection point. It's also the basis of the elimination method, which instead adds or subtracts multiples of the two equations to cancel one variable directly, often faster when neither equation is already solved for a single variable.