Leo+DadMade for Leo
Solving Linear Equations
Rung 4 of 4 · Mastery

Equations in the Real World

Real questions don't arrive as “2x + 3 = 11” — they arrive as a story about taxis or packs of cards. Turning words into an equation, solving it, then proving the answer: that's mastery.


BuildRead the story, see it become an equation, solve it — then watch the answer checked by substitution.
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A word problem is just an equation in a disguise. The skill is undressing it: decide what x stands for, build the equation from the sentence, then solve it the way you already know.

Words to Equation

Take: "A taxi charges a $3 flagfall plus $2 per kilometre, and the fare was $11 — how far was the trip?" Let x be the kilometres. The $2-per-km part is 2x, the flat $3 is + 3, and the total is 11: that's 2x + 3 = 11. Now it's the exact equation you solved on rung 2 — subtract 3, divide by 2, x = 4 km.

The move: name what x is, find the "per-something" amount (that's the multiplier), find the flat amount (that's the +/−), set it equal to the total — then solve as usual.

Always Check by Substitution

This is the bit that separates a right answer from a hoped-for one. Put your answer back into the original equation and see if it actually balances. For x = 4: 2 × 4 + 3 = 8 + 3 = 11 ✓. It matches the total, so 4 is genuinely correct — not just the number you got. The toy does this for you every time so the habit sticks.

Why This Is the Finish Line

The balance scale gave you the why; inverse operations gave you the how; watching the order and the signs made it safe. Reading a real situation, writing the equation, solving it, and verifying it — that's the whole loop, and it's exactly what equations are for. From here it carries straight into substitution and, before long, quadratics.

Us, Thinking Out Loud

What real cost in our week looks like “flat fee plus a rate”?

Why is substituting back worth the extra few seconds?

Of the four steps, which should we re-do in a fortnight?