Linear, simultaneous and quadratic equations
Solving an equation means finding the values that make it true. The method changes with the type, but the principle — do the same to both sides — never does.
Lesson overview
What you'll learn in this lesson
Manipulate algebraic expressions, solve equations and interpret graphs of functions.
Key learning points
- • Linear equations and inequalities
- • Simultaneous equations
- • Quadratic equations
This lesson at a glance
- 30 minutes
- 16 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 16
Watch first
Linear, simultaneous and quadratic equations
An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.
In a nutshell
Manipulate algebraic expressions, solve equations and interpret graphs of functions.
Learning cycle
Part 2 of 16
Learning cycle 1 of 2
Part 1 · Linear equations and inequalities
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 16
Learn
Linear equations and inequalities
Collect terms and use inverse operations. Inequalities work identically with one exception: multiplying or dividing by a negative reverses the sign. Represent the solution on a number line with an open circle for < or > and a filled circle for ≤ or ≥.
Explore the idea
Part 4 of 16
Learn
Simultaneous equations
Elimination: scale one or both equations so a variable has matching coefficients, then add or subtract. Substitution suits cases where one equation already gives a variable in terms of the other, and is essential when one equation is quadratic. Always substitute back to find the second variable and check both equations.
Quick check
Part 5 of 16
Quick check
Dividing an inequality by −2 means you must
Quick check
Part 6 of 16
Quick check
To solve x² − 5x + 6 = 0 by factorising, the brackets are
Reset break
Part 7 of 16
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 8 of 16
Learning cycle 2 of 2
Part 2 · Quadratic equations
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 16
Learn
Quadratic equations
Rearrange to the form ax² + bx + c = 0 first. Factorise where possible, since a product equals zero only when a bracket equals zero. Otherwise use the quadratic formula or complete the square. A negative discriminant means there are no real solutions, which is itself a valid answer.
Quick check
Part 10 of 16
Quick check
A quadratic with a negative discriminant has
Quick check
Part 11 of 16
Quick check
Before using the quadratic formula you must
Explore the idea
Part 12 of 16
Worked example
Worked answer: solve 3x + 2y = 16 and x − y = 2 (4 marks)
From the second equation, x = y + 2 (1). Substituting into the first: 3(y + 2) + 2y = 16, so 3y + 6 + 2y = 16 and 5y = 10 (1), giving y = 2 (1). Then x = 2 + 2 = 4 (1). Check in the first equation: 3(4) + 2(2) = 16, correct. Marks are lost most often by finding one variable and stopping.
Challenge round
Part 13 of 16
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 14 of 16
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
Elimination: scale one or both ____ so a variable has matching coefficients, then add or subtract.
Mastery quiz
Part 15 of 16
Marked quiz
End of lesson quiz: Linear, simultaneous and quadratic equations
4 questions, marked with the reasoning shown. No timer.
1. Dividing an inequality by −2 means you must
2. To solve x² − 5x + 6 = 0 by factorising, the brackets are
3. A quadratic with a negative discriminant has
4. Before using the quadratic formula you must
Lesson round-up
Part 16 of 16
Lesson round-up
Ready when you are
Quiz score
Not sat
Games
Not played
Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 16 · Watch & discover