Step by step
The question
Question: Two cars start 30 km apart and drive toward each other.
One drives 40 km/h faster than the other.
They meet after 1 hour.
How fast is each one?
1 Find what we know
What do we know?
They go toward each other.
One is 40 km/h faster than the other.
Both go for 1 hour.
At the start, they are 30 km apart.
What do we want?
The speed of each one.
2 Name the unknown
We know the faster speed from the slower one.
So we call the slower speed x km/h.
The other is 40 km/h faster: x + 40.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Car 1: x km/h, for 1 hour.
Car 2: x + 40 km/h, for 1 hour.
Distance = speed times time: 1x.
And for the other: 1(x + 40).
We keep x + 40 together in brackets.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is x + (x + 40) = 30.
Now we solve it.
Careful!
All of the speed x + 40 is times 1.
So it is 1(x + 40), not 1x + 40.
4 Open the brackets
A bracket with a plus or nothing in front can just be removed.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
x + x = 2x.
Now we have 2x + 40 = 30.
We want x by itself.
Two things are stuck to x: times 2 and + 40.
The + 40 is on the outside, so it comes off first.
Then the times 2.
6 Remove the + 40
To remove + 40, we take away 40.
We do it on both sides, so they stay equal.
+ 40 and − 40 make zero.
Cross them out.
30 − 40 = −10.
Now we have 2x = −10.
7 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
−10 divided by 2 is −5.
So x = −5!
8 Answer the question
x = −5 is not a speed.
So there is no answer.
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Last: answer the question.