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SAT: Linear equations in one variable

Choices and typed answers · SAT Math · 24 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Open the brackets
  2. Get x on one side
  3. Remove the + 1
  4. Check
  5. Pick the choice
  6. Why not the others

The steps for What value of x is the solution to the equation 2(x - 4) = 3x + 1?…. Every solution names its steps, so you always know what comes next.

Worked examples

3x + 5 = 20. What is the solution to the given equation? A) 5 B)… Easy

The question

Question: solve 3x + 5 = 20.

3x + 5 = 20

We want x by itself.

Two things are stuck to x: times 3 and + 5.

First x was times 3.

Then + 5.

To take them off, we go backwards.

First we remove the + 5.

Then the times 3.

x → × 3 → + 5
take off: − 5, then ÷ 3

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Remove the + 5

To remove + 5, we take away 5.

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 + 5 − 5 = 20 − 5

+ 5 and − 5 make zero.

Cross them out.

20 − 5 = 15.

20 − 5 = 15

Now we have 3x = 15.

3x = 15

2 Remove the 3

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

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

3x3 = 153

3 divided by 3 is 1.

So just x is left.

15 divided by 3 is 5.

153 = 5

So x = 5!

x = 5

3 Check

Let's check: put 5 in place of x.

Check: 3(5) + 5 = 15 + 5 = 20

We get 20, the same as the right side.

So x = 5 is right!

4 Pick the choice

Our answer is x = 5.

That is choice A.

A) 5

5 Why not the others

B is 25/3.

Wrong: added 5: 3x = 25.

B) 253 ✗ added 5: 3x = 25

C is 15.

Wrong: 3x = 15, but we still had to divide by 3.

C) 15 ✗ 3x = 15, but we still had to divide by 3

D is 45.

Wrong: 15 × 3: multiplied instead of dividing.

D) 45 ✗ 15 × 3: multiplied instead of dividing

Let's look back.

First: remove the + 5.

Then: remove the 3.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer
x = 5

▶ Watch it solved

2(x + 3) = 14. What value of x satisfies the given equation? A) 4 B)… Medium

The question

Question: solve 2(x + 3) = 14.

2(x + 3) = 14

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Remove the 2

The 2 multiplies the whole bracket.

Easy way: remove the 2 first.

Then we do not have to open the bracket.

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 = 142

2 divided by 2 is 1.

So just the bracket is left.

14 divided by 2 is 7.

142 = 7

Now we have x + 3 = 7.

We don't need the bracket any more.

x + 3 = 7

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 = 7 − 3

+ 3 and − 3 make zero.

Cross them out.

7 − 3 = 4.

7 − 3 = 4

So x = 4!

x = 4

3 Check

Let's check: put 4 in place of x.

Check: 2(4 + 3) = 2(7) = 14

We get 14, the same as the right side.

So x = 4 is right!

4 Pick the choice

Our answer is x = 4.

That is choice A.

A) 4

5 Why not the others

B is 5.5.

Wrong: 2x + 3 = 14: forgot 2 × 3.

B) 5.5 ✗ 2x + 3 = 14: forgot 2 × 3

C is 10.

Wrong: added 6: 2x = 20.

C) 10 ✗ added 6: 2x = 20

D is 11.

Wrong: x + 3 = 14: forgot the 2.

D) 11 ✗ x + 3 = 14: forgot the 2

Let's look back.

First: remove the 2.

Then: remove the + 3.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer
x = 4

▶ Watch it solved

What value of x is the solution to the equation 2(x - 4) = 3x + 1?… Harder

The question

Question: solve 2(x − 4) = 3x + 1.

2(x − 4) = 3x + 1

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Open the brackets

The 2 in front of the bracket multiplies every part inside.

So we write 2 times each part.

2 × x + 2 × (−4) = 3x + 1
− × + = −

Watch the signs.

2 times x is 2x.

2 times (−4) is −8.

2x − 8 = 3x + 1

2 Get x on one side

There is x on both sides.

We want x on one side only.

The numbers in front of x are 2 and 3.

We remove the smaller one, 2x.

Then the x part that is left is plus.

A plus x part is easier to work with.

So we remove the 2x from the left side.

To remove 2x, we take away 2x.

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.

2x − 8 − 2x = 3x + 1 − 2x

2x and − 2x make zero.

Cross them out.

3x − 2x = x.

3x − 2x = x

Now we have −8 = x + 1.

−8 = x + 1

Let's swap the two sides, so x is on the left.

Both sides are the same amount, so we can swap them.

x + 1 = −8

3 Remove the + 1

We want x by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

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

x + 1 − 1 = −8 − 1

+ 1 and − 1 make zero.

Cross them out.

−8 − 1 = −9.

−8 − 1 = −9

So x = −9!

x = −9

4 Check

Let's check: put −9 in place of x.

First the left side.

Left: 2(−9 − 4) = 2(−13) = −26

Now the right side.

Right: 3(−9) + 1 = −27 + 1 = −26

Both sides are −26.

So x = −9 is right!

Both sides equal −26

5 Pick the choice

Our answer is x = −9.

That is choice A.

A) −9

6 Why not the others

B is −5.

Wrong: 2x − 4 = 3x + 1: forgot 2 × 4.

B) −5 ✗ 2x − 4 = 3x + 1: forgot 2 × 4

C is 9/5.

Wrong: moved 3x without changing its sign: 5x = 9.

C) 95 ✗ moved 3x without changing its sign: 5x = 9

D is 9.

Wrong: lost the minus sign.

D) 9 ✗ lost the minus sign

Let's look back.

First: open the brackets.

Then: get x on one side.

Then: remove the + 1.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer
x = −9

▶ Watch it solved

Practice

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

Easy

  1. 3x + 5 = 20. What is the solution to the given equation? A) 5 B) 25/3 {added 5: 3x = 25} C) 15 {3x = 15, but we still had to divide by 3} D) 45 {15 × 3: multiplied instead of dividing}
    Answer
    x = 5
  2. If 4x - 7 = 21, what is the value of x? A) 3.5 {took away 7: 4x = 14} B) 7 C) 28 {4x = 28, but we still had to divide by 4} D) 112 {28 × 4: multiplied instead of dividing}
    Answer
    x = 7
  3. x/3 = 4. What value of x is the solution to the given equation? A) 1 {4 − 3: took away 3} B) 4/3 {4 ÷ 3: divided instead of multiplying} C) 7 {4 + 3: added 3} D) 12
    Answer
    x = 12
  4. What value of x is the solution to the equation 5x = -35? A) -175 {−35 × 5: multiplied instead of dividing} B) -40 {−35 − 5: took away 5} C) -7 D) 7 {lost the minus sign}
    Answer
    x = −7
  5. A plumber charges $40 plus $25 an hour. The bill was $140. How many hours was the job? A) 4 B) 5.6 {140 ÷ 25: forgot the $40} C) 7.2 {added 40: 180 ÷ 25} D) 100 {25x = 100, but we still had to divide by 25}
    Answer
    x = 4
  6. A taxi charges $4 plus $3 per mile. A ride costs $25. How many miles was the ride? A) 7 B) 25/3 {25 ÷ 3: forgot the $4} C) 29/3 {added 4: 29 ÷ 3} D) 21 {3x = 21, but we still had to divide by 3}
    Answer
    x = 7
  7. 2x + 9 = 3. What is the solution to the given equation? (type the answer)
    Answer
    x = −3
  8. A taxi charges $5 plus $2 per mile. A ride costs $21. How many miles was the ride? (type the answer)
    Answer
    x = 8

Medium

  1. 2(x + 3) = 14. What value of x satisfies the given equation? A) 4 B) 5.5 {2x + 3 = 14: forgot 2 × 3} C) 10 {added 6: 2x = 20} D) 11 {x + 3 = 14: forgot the 2}
    Answer
    x = 4
  2. 5x - 3 = 2x + 9. What is the solution to the given equation? A) 12/7 {added 2x: 7x = 12} B) 2 {took away 3: 3x = 6} C) 4 D) 12 {3x = 12, but we still had to divide by 3}
    Answer
    x = 4
  3. x/4 + 2 = 5. What is the solution to the given equation? A) 3/4 {x/4 = 3, then divided 3 by 4} B) 12 C) 18 {x + 2 = 20: multiplied only the 5 by 4} D) 28 {added 2: x/4 = 7}
    Answer
    x = 12
  4. 0.5x + 1.5 = 4. What value of x satisfies the given equation? A) 1.25 {4 − 1.5 = 2.5, then 2.5 × 0.5: multiplied instead of dividing} B) 2.5 {4 − 1.5 = 2.5, then forgot to divide by 0.5} C) 5 D) 11 {4 + 1.5 = 5.5: added 1.5 instead of taking it away}
    Answer
    x = 5
  5. 3(x - 2) = x + 8. What is the solution to the given equation? A) 1 {took away 6: 2x = 2} B) 3.5 {added x: 4x = 14} C) 5 {3x − 2 = x + 8: forgot 3 × 2} D) 7
    Answer
    x = 7
  6. A gym costs $20 to join and $10 a month. Tom paid $90. How many months did he pay for? A) 7 B) 9 {90 ÷ 10: forgot the $20} C) 11 {added 20: 110 ÷ 10} D) 70 {10x = 70, but we still had to divide by 10}
    Answer
    x = 7
  7. 2x/3 = 8. What is the solution to the given equation? (type the answer)
    Answer
    x = 12
  8. 4x + 3 = 2x + 8. What is the solution to the given equation? (type the answer)
    Answer
    x = 52
  9. An electrician charges $50 plus $30 an hour. The bill was $170. How many hours was the job? (type the answer)
    Answer
    x = 4
  10. 2(x + 3) = 2x + 5. How many solutions does the given equation have? A) zero B) exactly one {the x parts cancel: no single x is left to find} C) exactly two {a line-type equation never has exactly two} D) infinitely many {6 = 5 is never true: no number works}
    Answer
    6 = 5
  11. 3(x + 2) = 3x + 6. How many solutions does the given equation have? A) zero {6 = 6 is always true: every number works} B) exactly one {the x parts cancel: every x works, not one} C) exactly two {a line-type equation never has exactly two} D) infinitely many
    Answer
    6 = 6

Harder

  1. What value of x is the solution to the equation 2(x - 4) = 3x + 1? A) -9 B) -5 {2x − 4 = 3x + 1: forgot 2 × 4} C) 9/5 {moved 3x without changing its sign: 5x = 9} D) 9 {lost the minus sign}
    Answer
    x = −9
  2. If 3x + 6 = 21, what is the value of x + 2? A) 5 {that is x, not x + 2} B) 7 C) 15 {that is 3x, not x + 2} D) 21 {that is 3x + 6, not x + 2}
    Answer
    Careful: the question asks for x + 2, not x
  3. 3(x + 2) = 3x + k. In the given equation, k is a constant. The equation has infinitely many solutions. What is the value of k? (type the answer)
    Answer
    k = 6
  4. (x + 1)/3 = (x - 1)/2. What is the solution to the given equation? (type the answer)
    Answer
    x = 5
  5. 0.2(x + 5) = 0.5x - 1. What is the solution to the given equation? (type the answer)
    Answer
    x = 203