Steply

Changing the subject when it appears twice

Algebra 1 · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Clear the fraction
  2. Open the brackets
  3. Get the x terms on one side
  4. Take x out
  5. Remove the (y − 1)
  6. Check

The steps for Make x the subject of y = (x + 2)/(x - 3). Every solution names its steps, so you always know what comes next.

Worked examples

Ax = b + cx for x Easy

The question

Question: make x the subject of ax = b + cx.

ax = b + cx

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

cx is on the right.

To remove it, we take away cx.

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

ax − cx = b + cx − cx

Careful!

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

cx becomes −cx.

Careful: not ax + cx = b ✗

cx and −cx make zero.

Cross them out.

Now both x terms are on the left: ax − cx = b.

ax − cx = 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 − c) is b.

x(a − c) = b

3 Remove the (a − c)

(a − c)x means (a − c) times x.

We want just x.

To remove times (a − c), we divide by (a − c).

We do it on both sides.

(a − c)x(a − c) = b(a − c)

(a − c) divided by (a − c) is 1.

So just x is left.

So x = b/(a − c).

x = b(a − c)

4 Check

Let's check with numbers.

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

Our answer gives x = 4.

x = 43 − 2 = 41 = 4

Now we put all the numbers into the start.

First the left side.

Left: (3)(4) − (2)(4) = 12 − 8 = 4

Now the right side.

Right: 4

Both sides are 4.

So our answer is right!

Both sides equal 4

Let's look back.

First: get the x terms on one side.

Then: take x out.

Then: remove the (a − c).

Last: check.

Answer
x = b(a − c)

▶ Watch it solved

Make x the subject of a(x + b) = cx Medium

The question

Question: make x the subject of a(x + b) = cx.

a = cx

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

.

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

1 Open the brackets

What is in front of a bracket multiplies every part inside.

a times every part of (x + b).

ax + ab = cx

2 Get the x terms on one side

cx is on the right.

To remove it, we take away cx.

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

ax + ab − cx = cx − cx

Careful!

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

cx becomes −cx.

Careful: not ax + ab + cx = 0 ✗

cx and −cx make zero.

Cross them out.

Now we have ax + ab − cx = 0.

ax + ab − cx = 0

ab has no x.

To remove it from the left, we take away ab.

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

ax + ab − cx − ab = 0 − ab

ab and −ab make zero.

Cross them out.

Now both x terms are on the left: ax − cx = −ab.

ax − cx = −ab

3 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 − c) is −ab.

x(a − c) = −ab

4 Remove the (a − c)

(a − c)x means (a − c) times x.

We want just x.

To remove times (a − c), we divide by (a − c).

We do it on both sides.

(a − c)x(a − c) = −ab(a − c)

(a − c) divided by (a − c) is 1.

So just x is left.

So x = −ab over (a − c).

x = −ab(a − c)

5 Check

Let's check with numbers.

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

Our answer gives x = 3.

x = −(2)(3)2 − 4 = −6−2 = 3

Now we put all the numbers into the start.

First the left side.

Left: (2)(3) − (4)(3) = 6 − 12 = −6

Now the right side.

Right: −(2)(3) = −6

Both sides are −6.

So our answer is right!

Both sides equal −6

Let's look back.

First: open the brackets.

Then: get the x terms on one side.

Then: take x out.

Then: remove the (a − c).

Last: check.

Answer
x = −ab(a − c)

▶ Watch it solved

Make x the subject of y = (x + 2)/(x - 3) Harder

The question

Question: make x the subject of y = (x + 2)/(x − 3).

y = x + 2x − 3

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

.

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

1 Clear the fraction

The bottom, (x − 3), is in the way.

Times (x − 3) removes it.

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

y × (x − 3) = x + 2(x − 3) × (x − 3)

(x − 3) divided by (x − 3) is 1.

Only the top is left.

Now we have y(x − 3) = x + 2.

y(x − 3) = x + 2

2 Open the brackets

What is in front of a bracket multiplies every part inside.

y times every part of (x − 3).

xy − 3y = x + 2

3 Get the x terms on one side

x is on the right.

To remove it, we take away x.

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

xy − 3y − x = x + 2 − x

Careful!

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

x becomes −x.

Careful: not xy − 3y + x = 2 ✗

x and −x make zero.

Cross them out.

Now we have xy − 3y − x = 2.

xy − 3y − x = 2

−3y has no x.

To remove it from the left, we add 3y.

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

xy − 3y − x + 3y = 2 + 3y

−3y and 3y make zero.

Cross them out.

Now both x terms are on the left: xy − x = 2 + 3y.

xy − x = 2 + 3y

4 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 (y − 1) is 2 + 3y.

x(y − 1) = 2 + 3y

5 Remove the (y − 1)

(y − 1)x means (y − 1) times x.

We want just x.

To remove times (y − 1), we divide by (y − 1).

We do it on both sides.

(y − 1)x(y − 1) = 2 + 3y(y − 1)

(y − 1) divided by (y − 1) is 1.

So just x is left.

So x = (2 + 3y)/(y − 1).

x = 2 + 3y(y − 1)

6 Check

Let's check with numbers.

We pick y = 2.

Our answer gives x = 8.

x = 2 + 3(2)2 − 1 = 2 + 61 = 81 = 8

Now we put all the numbers into the start.

First the left side.

Left: (8)(2) − 8 = 16 − 8 = 8

Now the right side.

Right: 2 + 3(2) = 2 + 6 = 8

Both sides are 8.

So our answer is right!

Both sides equal 8

Let's look back.

First: clear the fraction.

Then: open the brackets.

Then: get the x terms on one side.

Then: take x out.

Then: remove the (y − 1).

Last: check.

Answer
x = 2 + 3y(y − 1)

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. ax = b + cx for x
    Answer
    x = b(a − c)
  2. make x the subject of ax + b = cx + d
    Answer
    x = d − b(a − c)
  3. make x the subject of 3x + a = bx
    Answer
    x = −a(3 − b)
  4. make x the subject of ax - 5 = 2x + b
    Answer
    x = b + 5(a − 2)
  5. make x the subject of px + q = rx
    Answer
    x = −q(p − r)
  6. make x the subject of 5x + c = dx + 2
    Answer
    x = 2 − c(5 − d)
  7. make x the subject of ax + 4 = x + b
    Answer
    x = b − 4(a − 1)

Medium

  1. make x the subject of a(x + b) = cx
    Answer
    x = −ab(a − c)
  2. make x the subject of p(x - q) = r(x + s)
    Answer
    x = rs + pq(p − r)
  3. make x the subject of 2(x + a) = b(x - 1)
    Answer
    x = −b − 2a(2 − b)
  4. make x the subject of a(x - 3) = 4x + b
    Answer
    x = b + 3a(a − 4)
  5. make x the subject of m(x + 2) = n(x - 5)
    Answer
    x = −5n − 2m(m − n)
  6. make x the subject of xy + 3 = 2x + y
    Answer
    x = y − 3(y − 2)
  7. make x the subject of ax - b = c - dx
    Answer
    x = c + b(a + d)

Harder

  1. make x the subject of y = (x + 2)/(x - 3)
    Answer
    x = 2 + 3y(y − 1)
  2. make x the subject of y = (2x + 1)/(x - 4)
    Answer
    x = 1 + 4y(y − 2)
  3. make x the subject of a = (x + b)/(x - c)
    Answer
    x = b + ac(a − 1)
  4. make x the subject of y = (3x - 2)/(x + 5)
    Answer
    x = −2 − 5y(y − 3)
  5. make x the subject of k = (x - 1)/(x + 2)
    Answer
    x = −1 − 2k(k − 1)
  6. make x the subject of y = (x + 4)/(x - 2)
    Answer
    x = 4 + 2y(y − 1)