GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher
Solving linear equations
An equation is a balance. Keep both sides equal and x has nowhere left to hide.
Your move
Four routes, one answer
Solve 3(2x + 11) = 9. Choose a first move, then a second, and see where you land. Then go back and take a different route.
First move → Second move
The 3 outside the bracket multiplies everything inside it. Whatever you do next, do it to BOTH sides, so the equation stays true.
2 × 2 = 4 routes to the answer, and the tree ends 4 times.
Every route keeps the two sides equal at every step. That's exactly why they all have to finish at the same value of x.
Maths · Algebra
Keep it balanced, line by line
Picture the = sign as the middle of a see-saw balance: whatever happens to one side must happen to the other, or it tips. Step forward and watch what is done to BOTH sides each time.
x is tangled up with a 7 and a 6. Peel them off one at a time, from the outside in.
Step 1 of 4
x is tangled up with a 7 and a 6. Peel them off one at a time, from the outside in.
Predict, then check
Warm-up, done for you: in 6x + 3 = 3x + 15, take 3x from both sides to get 3x + 3 = 15, so 3x = 12 and x = 4. Check: 6(4) + 3 = 27 and 3(4) + 15 = 27 ✓
Now a harder one. 5(x + 4) = −3(2x + 1) expands to 5x + 20 = −6x − 3. Which first move gets x onto one side AND keeps its coefficient positive?
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Keep both sides balanced, get x on its own, then check it.
What you need to know
- Do the same operation to both sides: add the inverse to remove a number, then divide by the coefficient (or multiply by its reciprocal) to leave x on its own.
- With a bracket you can expand first or divide both sides by the number outside the bracket; dividing first is often quicker when the other side divides exactly.
- Clear a fraction coefficient by multiplying both sides by its reciprocal, and clear a division by multiplying both sides by the divisor.
- With x on both sides, remove the smaller x-term so the coefficient of x stays positive.
- Check by substituting into the original equation; if the answer is not a whole number, leave it as a fraction in its simplest form.
The big picture
Solving an equation means finding the value of the unknown that makes both sides equal. You get the unknown on its own by doing the same thing to both sides: add the inverse to remove a number, and divide by the coefficient (the same as multiplying by its reciprocal) to leave x alone. Brackets, fractions and unknowns on both sides each have a sensible first move, but any order that keeps the two sides equal lands on the same answer. Finish by checking in the original equation and, in a problem, by answering what the question actually asked.
Key points
Worked example
Problem
Solve 7(x + 2) = 3(x − 6).
⚠ Watch out
Changing one side and forgetting the other. If you add −5 to the left, add −5 to the right as well: the moment the two sides are treated differently, the equation stops being true and every line after it is wrong.
Memory hook
Balance it, isolate it, check it, then answer it: the same to both sides, x on its own, back into the original, and one last look at what the question asked.
Check yourself
Solve 2(x + 9) = 4 twice: divide by 2 first, then expand first. Do both routes give the same x, and does it check out in the original equation?
Flashcards
(14)What does it mean to solve an equation?
What is the golden rule for keeping an equation true?
How do you remove the + 9 from 4x + 9 = 25?
How do you turn 4x into x?
How do you check a solution?
The answer isn't a whole number. How should you leave it?
What are the two ways to start an equation like 4(x + 1) = 20?
When is dividing by the number outside the bracket a good first move?
How do you clear a fraction coefficient, as in (2/7)(x + 1) = 6?
How do you clear a division, as in (x − 3)/4 = 5?
x is on both sides. Which x-term should you remove?
How do you solve an equation with brackets on both sides?
Does the order of your steps change the answer?
You've found x in a word problem. Are you finished?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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