Make x the subject of ax - b = c

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The question

Question: make x the subject of ax − b = c − dx.

ax − b = c − dx

The subject is the letter on its own: we want x = ...

Here x is in two places, so we must bring them together.

1 Get the x terms on one side

−dx is on the right.

To remove it, we add dx.

We do it on both sides, so they stay equal.

ax − b + dx = c ~~− dx + dx~~

Careful!

A term that moves to the other side changes its sign.

−dx becomes dx.

Careful: not ax − b − dx = c ✗

−dx and dx make zero.

Cross them out.

Now we have ax − b + dx = c.

ax − b + dx = c

−b has no x.

To remove it from the left, we add b.

We do it on both sides, so they stay equal.

ax ~~− b~~ + dx ~~+ b~~ = c + b

−b and b make zero.

Cross them out.

Now both x terms are on the left: ax + dx = c + b.

ax + dx = c + b

2 Take x out

x is in every part.

So we take it out once, and write what is left in a bracket.

ab + ac = a(b + c)

So x times (a + d) is c + b.

x(a + d) = c + b

3 Remove the (a + d)

(a + d)x means (a + d) times x.

We want just x.

To remove times (a + d), we divide by (a + d).

We do it on both sides.

~~(a + d)~~x~~(a + d)~~ = c + b(a + d)

(a + d) divided by (a + d) is 1.

So just x is left.

So x = (c + b)/(a + d).

x = c + b(a + d)

4 Check

Let's check with numbers.

We pick a = 2, b = 3, c = 4, d = 5.

Our answer gives x = 1.

x = 4 + 32 + 5 = 77 = 1

Now we put all the numbers into the start.

First the left side.

Left: (2)(1) + (5)(1) = 2 + 5 = 7

Now the right side.

Right: 4 + 3 = 7

Both sides are 7.

So our answer is right!

Both sides equal 7

Let's look back.

First: get the x terms on one side.

Then: take x out.

Then: remove the (a + d).

Last: check.

Answer