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Linear equations

Brackets, fractions, decimals, no solution, every number · Algebra 1 · 24 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Open the brackets
  2. Group like terms
  3. Get x on one side
  4. Remove the − 2
  5. Remove the 2
  6. Check

The steps for Solve 3(x - 2) + 4 = x + 10. Every solution names its steps, so you always know what comes next.

Worked examples

Solve 2(x + 3) = 10 Easy

The question

Question: solve 2(x + 3) = 10.

2(x + 3) = 10

1 Remove the 2

The 2 multiplies the whole bracket.

Easy way: remove the 2 first.

To remove times 2, we divide by 2.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

2(x + 3)2 = 102

2 divided by 2 is 1.

So just the bracket is left.

10 divided by 2 is 5.

102 = 5

Now we have x + 3 = 5.

We don't need the bracket any more.

x + 3 = 5

2 Remove the + 3

We want x by itself.

The + 3 is in the way.

To remove + 3, we take away 3.

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

x + 3 − 3 = 5 − 3

+ 3 and − 3 make zero.

Cross them out.

5 − 3 = 2.

5 − 3 = 2

So x = 2!

x = 2

3 Check

Let's check: put 2 in place of x.

Check: 2(2 + 3) = 2(5) = 10

We get 10, the same as the right side.

So x = 2 is right!

Let's look back.

First: remove the 2.

Then: remove the + 3.

Last: check.

Answer
x = 2

▶ Watch it solved

Solve 3(x - 2) + 4 = x + 10 Medium

The question

Question: solve 3(x − 2) + 4 = x + 10.

3(x − 2) + 4 = x + 10

1 Open the brackets

The 3 in front of the bracket multiplies every part inside.

So we write 3 times each part.

3 × x + 3 × (−2) + 4 = x + 10

Work out each multiplication.

3x − 6 + 4 = x + 10

2 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

Plain numbers go together.

−6 + 4 = −2.

−6 + 4 = −2

Now we have 3x − 2 = x + 10.

3x − 2 = x + 10

3 Get x on one side

There is x on both sides.

We want x on one side only.

So we remove the x from the right side.

To remove x, we take away x.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

3x − 2 − x = x + 10 − x

x and − x make zero.

Cross them out.

3x − x = 2x.

3x − x = 2x

Now we have 2x − 2 = 10.

2x − 2 = 10

We want x by itself.

Two things are stuck to x: times 2 and − 2.

The − 2 is on the outside, so it comes off first.

Then the times 2.

4 Remove the − 2

To remove − 2, we add 2.

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

2x − 2 + 2 = 10 + 2

− 2 and + 2 make zero.

Cross them out.

10 + 2 = 12.

10 + 2 = 12

Now we have 2x = 12.

2x = 12

5 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.

2x2 = 122

2 divided by 2 is 1.

So just x is left.

12 divided by 2 is 6.

122 = 6

So x = 6!

x = 6

6 Check

Let's check: put 6 in place of x.

First the left side.

Left: 3(6 − 2) + 4 = 3(4) + 4 = 12 + 4 = 16

Now the right side.

Right: 6 + 10 = 16

Both sides are 16.

So x = 6 is right!

Both sides equal 16

Let's look back.

First: open the brackets.

Then: group like terms.

Then: get x on one side.

Then: remove the − 2.

Then: remove the 2.

Last: check.

Answer
x = 6

▶ Watch it solved

Solve (3x - 1)/4 = (x + 3)/2 Harder

The question

Question: solve (3x − 1)/4 = (x + 3)/2.

Two fractions are equal.

3x − 14 = x + 32

1 Cross multiply

When two fractions are equal, we can cross multiply.

These two are equal.

(3x − 1) × 2 = (x + 3) × 4

Work out the multiplications.

2(3x − 1) = 4(x + 3)

2 Open the brackets

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

So we write each part times what is in front.

2 × 3x + 2 × (−1) = 4 × x + 4 × 3

Work out each multiplication.

6x − 2 = 4x + 12

3 Get x on one side

There is x on both sides.

We want x on one side only.

So we remove the 4x from the right side.

To remove 4x, we take away 4x.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

6x − 2 − 4x = 4x + 12 − 4x

4x and − 4x make zero.

Cross them out.

6x − 4x = 2x.

6x − 4x = 2x

Now we have 2x − 2 = 12.

2x − 2 = 12

We want x by itself.

Two things are stuck to x: times 2 and − 2.

The − 2 is on the outside, so it comes off first.

Then the times 2.

4 Remove the − 2

To remove − 2, we add 2.

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

2x − 2 + 2 = 12 + 2

− 2 and + 2 make zero.

Cross them out.

12 + 2 = 14.

12 + 2 = 14

Now we have 2x = 14.

2x = 14

5 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.

2x2 = 142

2 divided by 2 is 1.

So just x is left.

14 divided by 2 is 7.

142 = 7

So x = 7!

x = 7

6 Check

Let's check: put 7 in place of x.

First the left side.

Left: 3(7) − 14 = 21 − 14 = 204 = 5

Now the right side.

Right: 7 + 32 = 102 = 5

Both sides are 5.

So x = 7 is right!

Both sides equal 5

Let's look back.

First: cross multiply.

Then: open the brackets.

Then: get x on one side.

Then: remove the − 2.

Then: remove the 2.

Last: check.

Answer
x = 7

▶ Watch it solved

Practice

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

Easy

  1. 2(x + 3) = 10
    Answer
    x = 2
  2. 3(x - 4) = 9
    Answer
    x = 7
  3. 4(y + 1) = 8
    Answer
    y = 1
  4. -(x - 1) = 4
    Answer
    x = −3
  5. 5(n - 2) = 0
    Answer
    n = 2
  6. 0.5x = 3
    Answer
    x = 6
  7. x + 1.5 = 4
    Answer
    x = 2.5
  8. x/2 + x/3 = 10
    Answer
    x = 12
  9. 0.2x + 1 = 1.6
    Answer
    x = 3
  10. (x + 1)/2 = 3
    Answer
    x = 5

Medium

  1. 3(x - 2) + 4 = x + 10
    Answer
    x = 6
  2. 2(3x - 1) = 4x + 6
    Answer
    x = 4
  3. 5 - (x + 2) = 7
    Answer
    x = −4
  4. 4(x + 1) = 2(x - 3)
    Answer
    x = −5
  5. x/3 + 2 = x/2
    Answer
    x = 12
  6. 0.3x - 1.2 = 0.1x
    Answer
    x = 6
  7. (x + 1)/2 = x - 4
    Answer
    x = 9
  8. 2(x - 5) = 3(x + 1)
    Answer
    x = −13
  9. 2x + 1 = 2x + 3
    Answer
    1 = 3
  10. 3(x + 2) = 3x + 6
    Answer
    6 = 6

Harder

  1. (3x - 1)/4 = (x + 3)/2
    Answer
    x = 7
  2. 0.25(x - 4) = 1.5
    Answer
    x = 10
  3. 3 - 2(x - 1) = 4(2 - x)
    Answer
    x = 32
  4. 4(x - 1) - 2x = 2(x + 1)
    Answer
    −4 = 2