Comment résoudre les équations linéaires étape par étape : guide complet
Savoir résoudre les équations linéaires étape par étape est l'une des compétences les plus fondamentales en algèbre — et chaque équation linéaire cède au même processus en cinq étapes une fois que vous comprenez comment cela fonctionne. Une équation linéaire en une variable contient une inconnue (généralement x) élevée à la première puissance, et votre tâche est de trouver la valeur exacte qui rend l'équation vraie. Ce guide divise cette méthode en étapes claires et numérotées, puis vous guide à travers des exemples pratiqués à chaque niveau de difficulté que vous rencontrerez : équations à une et deux étapes, problèmes à plusieurs étapes avec distribution, équations avec des variables des deux côtés, équations avec des fractions et des problèmes contextuels du monde réel. Chaque exemple inclut l'étape de vérification — une habitude qui détecte les erreurs en secondes.
Sommaire
- 01Que signifie résoudre une équation linéaire étape par étape?
- 02Comment résoudre les équations linéaires étape par étape : la méthode des 5 étapes
- 03Comment appliquez-vous la méthode des 5 étapes aux équations à deux étapes et à plusieurs étapes?
- 04Quelle est la meilleure façon de résoudre les équations linéaires avec des fractions étape par étape?
- 05Comment traduisez-vous les problèmes contextuels en équations linéaires et les résolvez-vous?
- 06Quelles sont les erreurs les plus courantes lors de la résolution d'équations linéaires étape par étape?
- 07Questions fréquemment posées sur la résolution d'équations linéaires étape par étape
Que signifie résoudre une équation linéaire étape par étape?
Solving a linear equation means finding the unique value of the variable that makes the equation true. The phrase 'step by step' matters because you cannot jump straight to the answer — you must apply a sequence of inverse operations that gradually peel away everything surrounding x until it stands alone on one side of the equals sign. Each individual step follows two rules without exception: (1) only use inverse operations — the mathematical opposites of addition, subtraction, multiplication, and division — and (2) apply every operation to both sides simultaneously so the equation stays balanced. A linear equation in one variable takes the form ax + b = c, where a, b, and c are real-number constants and a ≠ 0. Unlike quadratic equations (which contain x²) or radical equations (which contain √x), a linear equation in one variable always produces exactly one solution — unless the variable terms cancel entirely, which signals either no solution or infinitely many solutions.
The goal of every step is the same: isolate x. Apply inverse operations to both sides equally until x stands alone with a coefficient of 1.
Comment appliquez-vous la méthode des 5 étapes aux équations à deux étapes et à plusieurs étapes?
Two-step equations require exactly two inverse operations to isolate x. Multi-step equations add distribution and like-term collection before those final operations. Work through each example below on your own before reading the solution — comparing your steps against the worked solution is the fastest way to identify gaps in your process.
1. Two-step: 5x + 8 = 38
Steps 1–3 do not apply (no parentheses, no like terms to combine, no x-term on the right). Step 4: Subtract 8 from both sides → 5x = 30. Divide both sides by 5 → x = 6. Step 5: Check: 5(6) + 8 = 30 + 8 = 38 ✓ Writing 'subtract 8 from both sides' explicitly — rather than crossing out the 8 mentally — builds the habit that prevents errors in harder problems.
2. Two-step: (x/3) − 4 = 2
Step 4: Add 4 to both sides → x/3 = 6. Multiply both sides by 3 → x = 18. Step 5: Check: 18/3 − 4 = 6 − 4 = 2 ✓ When x sits in the numerator of a fraction (x/3), treat the denominator as the operation — multiplication by 3 cancels the division. Do not divide by 3 again, which would produce x/9.
3. Multi-step with distribution: 3(2x − 1) + 7 = 28
Step 1: Distribute → 6x − 3 + 7 = 28. Step 2: Combine constants on the left → 6x + 4 = 28. Step 4: Subtract 4 → 6x = 24. Divide by 6 → x = 4. Step 5: Check: 3(2 × 4 − 1) + 7 = 3(7) + 7 = 21 + 7 = 28 ✓ The distribution happens first — students who skip to Step 4 early introduce an error that is hard to spot later.
4. Variables on both sides: 7x − 3 = 3x + 21
Step 3: Subtract 3x from both sides → 4x − 3 = 21. Step 4: Add 3 → 4x = 24. Divide by 4 → x = 6. Step 5: Check: 7(6) − 3 = 39; 3(6) + 21 = 39 ✓ Subtracting the smaller x-coefficient (3x) keeps the remaining coefficient positive (4x, not −4x), reducing the risk of a sign error in Step 4.
5. Multi-step with distribution on both sides: 4(x + 2) = 2(3x − 4) + 6
Step 1: Distribute both sides → 4x + 8 = 6x − 8 + 6 → 4x + 8 = 6x − 2. Step 3: Subtract 4x from both sides → 8 = 2x − 2. Step 4: Add 2 → 10 = 2x. Divide by 2 → x = 5. Step 5: Check: 4(5 + 2) = 28; 2(3 × 5 − 4) + 6 = 2(11) + 6 = 28 ✓
When x appears on both sides, move the smaller x-term across first. This keeps the remaining coefficient positive and makes the final division error-free.
Quelle est la meilleure façon de résoudre les équations linéaires avec des fractions étape par étape?
Fractions inside a linear equation are the most common source of calculation errors in algebra. The fix is the LCD (least common denominator) method: multiply every term in the equation by the LCD to clear all fractions in a single step. After that, you have a clean integer equation to solve normally. For decimal equations, multiply by a power of 10 — ×10 for one decimal place, ×100 for two — to achieve the same result.
1. Fractions with x in two terms: x/2 + x/5 = 7
The denominators are 2 and 5. LCD = 10. Multiply every term by 10: 10 × (x/2) + 10 × (x/5) = 10 × 7 5x + 2x = 70 7x = 70 x = 10. Check: 10/2 + 10/5 = 5 + 2 = 7 ✓ Multiplying by the LCD at the very start transforms a fraction equation into a straightforward integer equation in one move.
2. Fraction with grouped numerator: (3x + 1)/4 − x/2 = 3
LCD of 4 and 2 is 4. Multiply every term by 4: 4 × (3x + 1)/4 − 4 × (x/2) = 4 × 3 (3x + 1) − 2x = 12 x + 1 = 12 x = 11. Check: (3 × 11 + 1)/4 − 11/2 = 34/4 − 22/4 = 12/4 = 3 ✓ The numerator (3x + 1) acts as a single grouped term — the 4 in the LCD and the denominator 4 cancel, so do not try to distribute the 4 into the numerator separately.
3. Fractional coefficient: (5/6)x − 2 = 8
LCD = 6. Multiply every term by 6: 6 × (5/6)x − 6 × 2 = 6 × 8 5x − 12 = 48 5x = 60 x = 12. Check: (5/6)(12) − 2 = 10 − 2 = 8 ✓ Alternatively, add 2 first to get (5/6)x = 10, then multiply by the reciprocal 6/5: x = 12. Both routes give the same answer — use whichever is faster to set up.
4. Decimal equation: 0.6x − 1.2 = 3.6
Multiply every term by 10 to clear one-decimal-place values: 6x − 12 = 36 6x = 48 x = 8. Check: 0.6(8) − 1.2 = 4.8 − 1.2 = 3.6 ✓ For equations with two decimal places (such as 0.25x), multiply by 100 instead. The power of 10 you choose should eliminate all decimal points from every term simultaneously.
To clear fractions: multiply every term on both sides by the LCD. All fraction denominators cancel, leaving a clean integer equation to solve.
Comment traduisez-vous les problèmes contextuels en équations linéaires et les résolvez-vous?
Word problems test whether you can translate a real-world description into a linear equation and solve it. Follow this four-stage method every time: (1) identify the unknown and assign it a variable, (2) write one equation that captures every condition stated in the problem, (3) solve the equation using the 5-step method, (4) answer the original question in context and verify the solution makes sense.
1. Distance-rate-time: train travel
A train travels at 90 km/h. After how many hours will it have covered 360 km? Let h = number of hours. Equation: 90h = 360. Divide by 90 → h = 4 hours. Check: 90 × 4 = 360 ✓. The answer makes sense — 90 km/h for 4 hours gives exactly 360 km.
2. Savings goal: part-time earnings
Mia earns $18 per hour. She already has $126 saved and needs exactly $342 total. How many hours must she work? Let h = additional hours. Equation: 126 + 18h = 342. Subtract 126 → 18h = 216. Divide by 18 → h = 12 hours. Check: 126 + 18(12) = 126 + 216 = 342 ✓.
3. Consecutive integers
The sum of three consecutive integers is 87. Find all three. Let n = the smallest integer. The next two are n + 1 and n + 2. Equation: n + (n + 1) + (n + 2) = 87 3n + 3 = 87 3n = 84 n = 28. The integers are 28, 29, 30. Check: 28 + 29 + 30 = 87 ✓. Expressing consecutive integers as n, n + 1, n + 2 automatically captures their relationship without needing a second variable.
4. Geometry: rectangle perimeter
A rectangle is 4 m longer than it is wide. Its perimeter is 56 m. Find the width and length. Let w = width. Then length = w + 4. Perimeter: 2(length + width) = 56 2(w + 4 + w) = 56 2(2w + 4) = 56 4w + 8 = 56 4w = 48 w = 12 m; length = 16 m. Check: 2(16 + 12) = 2(28) = 56 ✓.
Word problem steps: (1) name the unknown. (2) write one equation from the problem's conditions. (3) solve. (4) verify the answer makes sense in the real-world context.
Quelles sont les erreurs les plus courantes lors de la résolution d'équations linéaires étape par étape?
These errors appear in student work at every level of algebra. Recognizing them before you encounter them in your own working is far more effective than discovering them in marked assignments.
1. Distributing to only the first term inside parentheses
In 5(x − 4), students often write 5x − 4 instead of 5x − 20. The factor outside must multiply every term inside. With a negative multiplier: −3(x − 7) = −3x + 21, not −3x − 21. The negative distributes to both x and −7, so −3 × (−7) = +21. Always check the sign of each product individually.
2. Applying an inverse operation to only one side
In 4x + 9 = 25, subtracting 9 from only the left gives 4x = 25 — wrong. You must subtract 9 from both sides: 4x = 16, so x = 4. Writing the operation beneath both sides before simplifying makes the requirement visual and prevents this error.
3. Sign error when dividing by a negative coefficient
In −6x = 30, dividing both sides by −6 gives x = −5, not x = 5. A positive divided by a negative is negative: 30 ÷ (−6) = −5. Always verify by substituting: −6 × (−5) = 30 ✓. If you prefer, flip both signs first (multiply both sides by −1) to get 6x = −30, then divide by 6: x = −5.
4. Combining unlike terms
3x and 7 cannot be combined — one is a variable term and the other is a constant. Similarly, 4x and 4x² are unlike because the exponents differ. Only terms with identical variable parts can be combined. A common mistake is writing 3x + 7 = 10x when trying to simplify both sides at once.
5. Checking a simplified version instead of the original equation
Always substitute your answer back into the original equation, not a version you simplified partway through. A simplification error might produce a wrong equation that your answer satisfies — but the original catches the mistake immediately. For example, if you mistakenly simplified 2x + 3 = 11 to 2x = 13, your answer x = 6.5 checks out in the wrong equation but fails the original.
Questions fréquemment posées sur la résolution d'équations linéaires étape par étape
These are the questions students ask most often when working through how to solve linear equations step by step for the first time.
1. What is the very first step when solving any linear equation?
Look for parentheses. If any exist, distribute first. If there are none, look for fractions and clear them by multiplying every term by the LCD. If neither applies, collect all x-terms on one side and all constants on the other, then isolate x using inverse operations. Starting with distribution and fraction-clearing prevents the cascade of errors that comes from trying to isolate x while grouped or fractional terms remain.
2. Why must I apply every operation to both sides?
An equation is a statement of equality. Both sides represent the same quantity. Applying an operation to only one side changes that quantity on that side alone, breaking the equality and producing a different equation whose solution may not match the original. Think of a balance scale: adding weight to one side without adding the same to the other causes it to tip.
3. Can a linear equation have no solution or infinite solutions?
Yes. If all x-terms cancel and leave a false statement (such as 3 = 8), no value of x satisfies the equation — the answer is 'no solution.' If they cancel and leave a true statement (such as 5 = 5), every real number is a solution — the answer is 'all real numbers' or 'infinite solutions.' These results look like errors at first, but they are valid outcomes of the 5-step method applied correctly.
4. How do I know when to use the LCD method versus just dividing?
Multiply by the LCD when the equation contains fractions with different denominators or when x appears inside a fraction's numerator alongside a constant (like (2x + 3)/5). Divide when x has a simple integer coefficient and no fractions are present — for example, in 4x = 28 simply divide both sides by 4. The LCD method is the more general strategy and works in all cases, so using it consistently avoids having to choose.
5. How long does it take to get fast at solving linear equations step by step?
Most students reach reliable speed within two or three focused practice sessions covering each equation type in sequence: one-step, two-step, multi-step, fractions, and word problems. Follow the 5-step checklist strictly at first even when steps seem unnecessary, until the sequence becomes automatic. Skipping steps to save time early on creates habits that slow you down on complex problems later.
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Comment résoudre les équations linéaires étape par étape : la méthode des 5 étapes
This five-step sequence applies to every linear equation you will encounter in algebra. Use it as a checklist — work through steps 1 to 5 in order rather than skipping ahead based on what seems obvious. Skipping steps is the leading cause of arithmetic errors on algebra tests.
1. Step 1: Distribute across parentheses
If any terms are grouped in parentheses, expand them using the distributive property before anything else. In 3(x + 4) = 21, distribute first: 3x + 12 = 21. Pay close attention to negative multipliers: −2(x − 5) = −2x + 10, not −2x − 10. The negative must multiply every term inside. Distributing incorrectly is the single most common source of errors in multi-step linear equations.
2. Step 2: Combine like terms on each side
After distributing, look at each side separately and combine terms with identical variable parts. On the left side of 5x − 2x + 9 = 3, combine 5x − 2x = 3x, leaving 3x + 9 = 3. You cannot combine a variable term with a constant — 5x + 3 does not simplify further. Always simplify each side before moving anything across the equals sign.
3. Step 3: Move all variable terms to one side
If x appears on both sides, use addition or subtraction to collect all variable terms on one side and all constants on the other. For 5x + 6 = 2x + 18, subtract 2x from both sides: 3x + 6 = 18. Prefer moving the smaller x-term — this keeps the remaining coefficient positive and prevents a later sign error when dividing.
4. Step 4: Isolate x using inverse operations
With x on one side and constants on the other, apply inverse operations to reduce the equation to x = [number]. For 3x + 6 = 18, subtract 6 from both sides: 3x = 12, then divide both sides by 3: x = 4. Work in reverse order of operations — undo addition and subtraction before undoing multiplication and division.
5. Step 5: Check your answer in the original equation
Substitute your solution back into the original equation — not a simplified version, the original. Evaluate both sides completely. If they match, the answer is correct. For x = 4 in 5x + 6 = 2x + 18: left = 5(4) + 6 = 26; right = 2(4) + 18 = 26 ✓. This check takes ten seconds and catches the vast majority of arithmetic mistakes before they cost marks.