Year 8AlgebraEquations and relationshipsYear 8 Maths

Equations, and what the letter is doing

What this lesson covers

  • A variable can be used to represent: an unknown number, often in formulae; a quantity that can vary or change; a specific unknown value to be solved.
  • The solution to an equation satisfies that equation.
  • Equations can be rearranged using inverse operations (e.g. addition and subtraction, multiplication and division).
  • Solutions to equations can be checked using substitution.
  • Equations can be solved through trial and error, but this can be an inefficient method.
Key idea

A letter does three different jobs

  • In 3a = 18, the a is one unknown number. Find it.
  • In A = bh, the letters are a formula. Put numbers in.
  • In y = 3x + 4, they are a relationship. Neither is fixed.
Looks likeThe letter isSo you
3a = 18one unknown valuesolve for it
A = bha formulasubstitute into it
y = 3x + 4a relationshipexplore it
Key idea

Why that distinction matters

  • Asked to "solve" A = bh, students go looking for b.
  • There is nothing to find. It is a formula, not a puzzle.
  • Read which job the letter is doing before you start.
Looks likeThe letter isSo you
3a = 18one unknown valuesolve for it
A = bha formulasubstitute into it
y = 3x + 4a relationshipexplore it
Key idea

What solving actually means

  • 3a = 18. The solution is the value that makes it true.
  • a = 6, because 3 × 6 really is 18.
  • Any other number makes the statement false.
What solving actually means
Key idea

Think of it as a balance

  • The two sides weigh the same. That is what = means.
  • Take 5 off one pan and it tips, unless you take 5 off both.
  • Every legal move is one you do to both sides.
Think of it as a balance
Key idea

Undo with the opposite operation

  • Something added is undone by subtracting.
  • Something multiplied is undone by dividing.
  • These pairs are called inverse operations.
If it wasDo this
Added 7undo bysubtracting 7
Subtracted 3undo byadding 3
Multiplied by 5undo bydividing by 5
Divided by 2undo bymultiplying by 2
Worked example

Solving 2x + 5 = 17

  • The 5 was added last, so take it off both sides first.
  • That leaves 2x = 12.
  • The x was doubled, so halve both sides. x = 6.
LineMoveGives
2x + 5 = 17take 5 off both pans2x = 12
2x = 12halve both pansx = 6
Check2 × 6 + 517
Worked example

Could you just guess instead?

  • Try 5s + 9 = −18 by trial. s = −5 gives −16. Close.
  • s = −6 gives −21. Too far. s = −5.5 gives −18.5.
  • Three goes, still not there. And the answer is not tidy.
GuessWorkingGives
s = −55(−5) + 9−16
s = −65(−6) + 9−21
s = −5.55(−5.5) + 9−18.5
Target−18
Key idea

In fairness, guessing is sometimes fine

  • For 3a = 18 you saw the answer instantly. No method needed.
  • Guessing is quick when the answer is small and whole.
  • It collapses when it is negative, or a decimal, or both.
EquationGuessingVerdict
3a = 18instantfine
5s + 9 = −18noundo it instead
Worked example

So undo it instead

  • 5s + 9 = −18. Subtract 9 from both sides: 5s = −27.
  • Divide both sides by 5: s = −5.4.
  • One pass, no guessing, and it works for any numbers.
LineMoveGives
5s + 9 = −18subtract 9 from both5s = −27
5s = −27divide both by 5s = −5.4
Key idea

Notice the scales have gone

  • A pan cannot hold −27 kilograms. The picture has run out.
  • The algebra has not. It handles negatives without blinking.
  • The balance explained why the method is allowed. It is done now.
BalanceAlgebra
2x + 5 = 17yesyes
5s + 9 = −18noyes
t + 7 = 6.5noyes
Worked example

One more the scales cannot hold

  • t + 7 = 6.5. Something plus 7 gives less than 7.
  • So t must be negative. Subtract 7 from both sides.
  • t = −0.5. Predicted before we calculated it.
LineMoveGives
t + 7 = 6.5subtract 7 from botht = −0.5
Check−0.5 + 76.5
Notethe answer was smaller than 7
Key idea

You can always check your own answer

  • Put your answer back into the original equation.
  • 5 × −5.4 is −27, and −27 + 9 is −18. It works.
  • No answer key needed. Algebra marks itself.
LineMoveGives
5s + 9 = −18subtract 9 from both5s = −27
5s = −27divide both by 5s = −5.4
Check5(−5.4) + 9−18
Worked example

Substituting, which is not solving

  • Find w + 12 when w = 4. Nothing to solve. Put 4 in.
  • 16. Same for y = 3x + 4 at x = 2, which gives 10.
  • And A = bh with b = 5 and h = 4 gives 20.
GivenWorkingAnswer
w + 12, w = 44 + 1216
y = 3x + 4, x = 23(2) + 410
A = bh, b = 5, h = 45 × 420
Key idea

Now: when it is not equal, but at least

  • Some questions ask for a range, not a single value.
  • "At least 5" is not one number, it is all of them from 5 up.
  • We write that with an inequality sign.
SignMeansEnd point
>greater thannot including it
≥greater than or equal toincluding it
<less thannot including it
≤less than or equal toincluding it
Worked example

Solving one works the same way

  • t − 3 ≥ −5. Add 3 to both sides, exactly as before.
  • t ≥ −2.
  • So t can be −2, or anything above it.
LineMoveGives
t − 3 ≥ −5add 3 to both sidest ≥ −2
So t can be−2, −1, 0, 1, ...
And −2 isincluded
Key idea

And the circle tells you about the end

  • t ≥ −2 includes −2, so the circle is filled in.
  • If it were t > −2, the circle would be hollow.
  • One stroke in the notation. Very visible on a line.
And the circle tells you about the end
Your turn

Your turn: four questions

  • Pause here and try all four.
  • Question 3 is a substitution, not an equation. Do not solve it.
  • Check every answer by putting it back in.
Question
1Solve x + 8 = 3.
2Solve 4m − 3 = 9.
3Find 2p + 7 when p = 5.
4Solve y − 4 ≤ 1, and say whether 5 is included.
Show answer

1. x = −5. Subtract 8 from both sides. The answer is negative, which is fine. 2. m = 3. Add 3 to both sides to get 4m = 12, then divide by 4. 3. 17. Substitute, do not solve: 2(5) + 7. 4. y ≤ 5, and yes, 5 is included, because the symbol has the line under it. On a number line that is a filled circle.

Worked example

The answers

  • x = −5, from subtracting 8. A negative answer is fine.
  • m = 3, and 2p + 7 at p = 5 is 17.
  • y ≤ 5, and 5 is included, so the circle is filled.
Answer
1x = −5
2m = 3
317
4y ≤ 5, and 5 counts
Summary

What to take away

  • Read what the letter is doing before you do anything to it.
  • Undo the operations in reverse, doing the same to both sides.
  • Then substitute your answer back, every single time.
Remember
A letterdoes three jobs; read which one
Solvingundo the operations, in reverse
Both sidesalways do the same to each
Checkingsubstitute your answer back in
≥ and >differ over whether the end point counts

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