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Literal equations

Rearrange a formula · Algebra 1 · 23 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. w on the left
  2. Remove the 2l
  3. Remove the 2
  4. Check

The steps for Solve P = 2l + 2w for w. Every solution names its steps, so you always know what comes next.

Worked examples

Solve A = lw for w Easy

The question

Question: solve for w in A = lw.

We treat the other letters like numbers.

A = lw

1 w on the left

w is on the right.

We swap the two sides, so w is on the left.

They are still equal.

lw = A

2 Remove the l

lw means l times w.

We want just w.

To remove times l, we divide by l.

We do it on both sides.

lwl = Al

l divided by l is 1.

So just w is left.

So w = A over l.

w = Al

3 Check

Let's check with numbers.

We pick A = 4, l = 2.

Our answer gives w = 2.

w = 42 = 2

Now we put all the numbers into the start.

First the left side.

Left: 4

Now the right side.

Right: (2)(2) = 4

Both sides are 4.

So our answer is right!

Both sides equal 4

Let's look back.

First: w on the left.

Then: remove the l.

Last: check.

Answer
w = Al

▶ Watch it solved

Solve P = 2l + 2w for w Medium

The question

Question: solve for w in P = 2l + 2w.

We treat the other letters like numbers.

P = 2l + 2w

1 w on the left

w is on the right.

We swap the two sides, so w is on the left.

They are still equal.

2l + 2w = P

2 Remove the 2l

We want w by itself.

The 2l is in the way.

To remove 2l, we take away 2l.

We do it on both sides.

2l + 2w − 2l = P − 2l

They make zero.

Cross them out.

Now we have 2w = P − 2l.

2w = P − 2l

3 Remove the 2

2w means 2 times w.

We want just w.

To remove times 2, we divide by 2.

We do it on both sides.

2w2 = P − 2l2

2 divided by 2 is 1.

So just w is left.

So w = (P − 2l)/2.

w = P − 2l2

4 Check

Let's check with numbers.

We pick P = 6, l = 2.

Our answer gives w = 1.

w = 6 − 2(2)2 = 6 − 42 = 22 = 1

Now we put all the numbers into the start.

First the left side.

Left: 6

Now the right side.

Right: 2(2) + 2(1) = 4 + 2 = 6

Both sides are 6.

So our answer is right!

Both sides equal 6

Let's look back.

First: w on the left.

Then: remove the 2l.

Then: remove the 2.

Last: check.

Answer
w = P − 2l2

▶ Watch it solved

Ax + bx = c for x Harder

The question

Question: solve for x in ax + bx = c.

We treat the other letters like numbers.

ax + bx = c

1 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 + b) is c.

x(a + b) = c

2 Remove the (a + b)

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

We want just x.

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

We do it on both sides.

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

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

So just x is left.

So x = c/(a + b).

x = c(a + b)

3 Check

Let's check with numbers.

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

Our answer gives x = 1.

x = 52 + 3 = 55 = 1

Now we put all the numbers into the start.

First the left side.

Left: (2)(1) + (3)(1) = 2 + 3 = 5

Now the right side.

Right: 5

Both sides are 5.

So our answer is right!

Both sides equal 5

Let's look back.

First: take x out.

Then: remove the (a + b).

Last: check.

Answer
x = c(a + b)

▶ Watch it solved

Practice

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

Easy

  1. A = lw for w
    Answer
    w = Al
  2. A = bh for b
    Answer
    b = Ah
  3. d = rt for t
    Answer
    t = dr
  4. y = x + b for b
    Answer
    b = y − x
  5. V = lwh for h
    Answer
    h = Vlw
  6. W = Fd for d
    Answer
    d = WF

Medium

  1. P = 2l + 2w for w
    Answer
    w = P − 2l2
  2. y = mx + b for x
    Answer
    x = y − bm
  3. A = (1/2)bh for b
    Answer
    b = 2Ah
  4. V = (1/3)Bh for h
    Answer
    h = 3VB
  5. I = Prt for r
    Answer
    r = IPt
  6. F = ma for a
    Answer
    a = Fm
  7. v = u + at for t
    Answer
    t = v − ua
  8. P = a + b + c for c
    Answer
    c = P − a − b

Harder

  1. ax + bx = c for x
    Answer
    x = c(a + b)
  2. A = P + Prt for t
    Answer
    t = A − PPr
  3. 3x + 2y = 12 for y
    Answer
    y = 12 − 3x2
  4. ax - b = c for x
    Answer
    x = c + ba
  5. E = mc^2 for m
    Answer
    m = Ec2
  6. T = 2a + 3b for b
    Answer
    b = T − 2a3
  7. P = 2(l + w) for l
    Answer
    l = P2 − w
  8. A = (a + b)h/2 for h
    Answer
    h = 2A(a + b)
  9. 2x - 2y = 8 for y
    Answer
    y = x − 4