System of 2 Equations with 2 Unknowns

System of 2 Equations with 2 Unknowns. Practice questions to deepen understanding of systems of 2 equations with 2 unknowns. Online math practice with full solutions and step-by-step explanations.

System of 2 equations practice — 40 questions: substitution, addition/subtraction, special cases (no/infinite solutions), word problems. What a system of equations is, what a solution is, verifying a solution, three solution methods, when to use substitution / when to use addition.

40 questions

Question 1
2.50 pts

🔢 :
2 2 ?

Explanation:

💡 Detailed Explanation:

A system of equations — like a puzzle with two clues! 🔍

Real-life example:

🛒 You bought apples and bananas.
Clue 1: Total 10 items
Clue 2: Apples cost 3× bananas

Two clues → we need both to find the answer!
Question 2
2.50 pts

🔢 :
system of equations?

Explanation:

💡 Detailed Explanation:

What is the solution? 🎯

Both equations satisfied!
Question 3
2.50 pts

🔢 :
x=2, y=3 ?

x + y = 5
x - y = -1

Explanation:

💡 Detailed Explanation:

Substitution — always substitute! 📝

given: x=2, y=3

Step 1:
x + y = 5
2 + 3 = 5 ✓

Step 2:
x - y = -1
2 - 3 = -1 ✓

Conclusion:
→ Yes, !

⭐ Conclusion: always !

Answer: Yes — it is a solution!

Question 4
2.50 pts

🔢 Verification:
Is x=4, y=1 a solution to the system?

x + y = 5
2x + y = 7

Explanation:

💡 Detailed explanation:

Careful verification 📝

Given: x=4, y=1

Check equation 1:
x + y = 5
4 + 1 = 5 ✓

Check equation 2:
2x + y = 7
2(4) + 1 = 8 + 1 = 9 ≠ 7 ✗

Conclusion:
Satisfies only one equation!
→ Not a solution to the system

⚠️ Must satisfy both!

Correct answer: No — satisfies only the first equation

Question 5
2.50 pts

🔢 :
system of equations?

Explanation:

💡 Detailed Explanation:

Solution 📐

Question 6
2.50 pts

🔢 Strategy:
When is it best to use the substitution method?

Explanation:

💡 Detailed explanation:

When to substitute? 🎯

Substitution is suitable when:

✅ A variable is already isolated:
y = 3x + 1
2x + y = 7

✅ Coefficient 1 (easy to isolate):
x + 2y = 5 → x = 5 - 2y
3x + y = 8

⚠️ Less convenient when:
3x + 4y = 10
2x + 5y = 8
(produces ugly fractions)

Correct answer: When one of the variables is already isolated or easy to isolate (coefficient 1)

Question 7
2.50 pts

🔢 :
/?

Explanation:

💡 Detailed Explanation:

When to use the elimination method? 🎯

Use it when:

✅ Opposite coefficients:
2x + 3y = 10
2x − 3y = 4
→ Add the equations to eliminate x

✅ Equal coefficients:
x + 2y = 7
x + y = 5
→ Subtract to eliminate x
Question 8
2.50 pts

🔢 :
system of equations?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

x
Question 9
2.50 pts

🔢 :
:

y = 2
x + y = 5

Explanation:

💡 Detailed Explanation:

Simplest substitution! 📝

y is already known!

y = 2

Substitute into the second equation:
x + y = 5
x + 2 = 5
x = 3

✅ Answer: x = 3, y = 2
Question 10
2.50 pts

🔢 :
:

x = 4
2x + y = 11

Explanation:

💡 Detailed Explanation:

Direct substitution 📝

x is known: x = 4

Substitute:
2x + y = 11
2·4 + y = 11
8 + y = 11
y = 3

✅ Answer: x = 4, y = 3
Question 11
2.50 pts

🔢 :
:

y = x + 1
x + y = 7

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Question 12
2.50 pts

🔢 :
:

x + y = 6
2x + 3y = 14

Explanation:

💡 Detailed Explanation:

Solution 📐

Step 1: Analysis 🔍 x

x + y = 6
x = 6 - y

Step 2: Calculation 📝

2x + 3y = 14
2(6-y) + 3y = 14
12 - 2y + 3y = 14
12 + y = 14
y = 2

Step 3: Result 📝 -x

x = 6 - 2 =
Question 13
2.50 pts

🔢 Substitution method:
Solve:

y = 2x
3x + y = 10

Explanation:

💡 Detailed explanation:

Substitution with an expression 📝

y is isolated: y = 2x

Substitute in the second equation:

3x + y = 10
3x + 2x = 10
5x = 10
x = 2

Find y:

y = 2x = 2·2 = 4

Check: 3(2) + 4 = 6 + 4 = 10 ✓

Correct answer: x = 2, y = 4

Question 14
2.50 pts

🔢 :
:

x = y - 3
2x + y = 9

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Isolate x: x = y - 3

Substitute:

2x + y = 9
2(y-3) + y = 9
2y - 6 + y = 9
3y - 6 = 9
3y = 15
y = 5

Find x:

x = y - 3 = 5 - 3 =
Question 15
2.50 pts

🔢 :
:

x - y = 2
3x + 2y = 16

Explanation:

💡 Detailed Explanation:

Solution 📐

Details: x ( 1)

Step 1: Understanding the Problem 🔍
x - y = 2
x = y + 2

Step 2: Calculation 📝
3x + 2y = 16
3(y+2) + 2y = 16
3y + 6 + 2y = 16
5y = 10
y = 2

3:
x =
Question 16
2.50 pts

🔢 :
:

y = 3x - 5
2x + y = 10

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Isolate y: y = 3x - 5

Substitute:

2x + y = 10
2x + (3x-5) = 10
2x + 3x - 5 = 10
5x - 5 = 10
5x = 15
x = 3

Find y:
y = 3(3) - 5 = 9 - 5 =
Question 17
2.50 pts

🔢 :
:

x + y = 8
x + 2y = 11

Explanation:

💡 Detailed Explanation:

- x equal! 📝

Details:
x + y = 8 (1)
x + 2y = 11 (2)

x equal ( 1)

: (2) - (1)

(x + 2y) - (x + y) = 11 - 8
x + 2y - x - y = 3
y = 3

Step 1:
x + 3 = 8
Question 18
2.50 pts

🔢 :
:

x + y = 7
x - y = 3

Explanation:

💡 Detailed Explanation:

- y because! 📝

- y !
Question 19
2.50 pts

🔢 Elimination method:
Solve:

2x + 3y = 13
2x - 3y = -1

Explanation:

💡 Detailed explanation:

Opposite y coefficients: +3 and -3 📝

Add the equations:

(2x + 3y) + (2x - 3y) = 13 + (-1)
2x + 3y + 2x - 3y = 12
4x = 12
x = 3

Substitute in the first equation:

2(3) + 3y = 13
6 + 3y = 13
3y = 7
y = 7/3

Correct answer: x = 3, y = 7/3

Question 20
2.50 pts

🔢 :
:

3x + 2y = 12
3x + 5y = 21

Explanation:

💡 Detailed Explanation:

- x equal (3) 📝

x equal!

3x + 2y = 12 (1)
3x + 5y = 21 (2)

: (2) - (1)

(3x + 5y) - (3x + 2y) = 21 - 12
3x + 5y - 3x - 2y = 9
3y = 9
y = 3

Conclusion:
3x + 2(3) = 12
3x + 6 = 12
3x = 6
Question 21
2.50 pts

🔢 :
:

x + y = 5
2x + 3y = 13

Explanation:

💡 Detailed Explanation:

Need to multiply first! 📝

Problem: coefficients not equal/opposite

Solution: Multiply equation 1 by 2
x + y = 5 → 2x + 2y = 10
Now subtract from 2x + y = 7:
2x + 2y = 10
−(2x + y = 7)
y = 3

✅ Substitute back: x = 2, y = 3
Question 22
2.50 pts

🔢 :
:

2x + 3y = 8
3x + 2y = 7

Explanation:

💡 Detailed Explanation:

Solution 📐

equal x:

1 × 3: 6x + 9y = 24
2 × 2: 6x + 4y = 14

Conclusion:
(6x + 9y) - (6x + 4y) = 24 - 14
5y = 10
y = 2

Conclusion:
2x + 3(2) = 8
2x + 6 = 8
2x = 2
Question 23
2.50 pts

🔢 :
:

x + 2y = 7
3x + 4y = 17

Explanation:

💡 Detailed Explanation:

Multiply to create opposite coefficients 📝

Match y coefficients:

Equation 1 × 2: 2x + 4y = 14
Equation 2: 2x + 3y = 12
Subtract: y = 2

Substitute: 2x + 4(2) = 14 → 2x = 6 → x = 3

✅ Answer: x = 3, y = 2
Question 24
2.50 pts

🔢 :
:

4x - 3y = 11
2x + 5y = 25

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

2 -2:

4x - 3y = 11
4x + 10y = 50

Conclusion:
(4x + 10y) - (4x - 3y) = 50 - 11
4x + 10y - 4x + 3y = 39
13y = 39
y = 3

Conclusion:
2x + 5(3) = 25
2x + 15 = 25
2x = 10
Question 25
2.50 pts

🔢 :
?

x + y = 5
x + y = 7

Explanation:

💡 Detailed Explanation:

! 📊

!
Question 26
2.50 pts

🔢 :
?

x + y = 5
2x + 2y = 10

Explanation:

💡 Detailed Explanation:

! 📊

Details:

2 = 1 × 2

2x + 2y = 10
÷2: x + y = 5

!

: Solution 📐

Yes :
(0,5), (1,4), (2,3), (3,2), ...

!

: -

Question 27
2.50 pts

🔢 :
?

2x - 4y = 6
x - 2y = 5

Explanation:

💡 Detailed Explanation:

Solution 📐

1 -2:

2x - 4y = 6
÷2: x - 2y = 3

Step 2:
x - 2y = 3 ( 1)
x - 2y = 5 ( 2)

!
equal -3 -5?
!

→

: -

Question 28
2.50 pts

🔢 :
system of equations?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Question 29
2.50 pts

📝 :
20.
4.

find .

Explanation:

💡 Detailed Explanation:

No 📝

Details:
x = large
y = small

Conclusion:
" 20": x + y = 20
" 4": x - y = 4

():
(x + y) + (x - y) = 20 + 4
2x = 24
x = 12

y = 20 - 12 = 8

Conclusion: 12+8=20 ✓,
Question 30
2.50 pts

📝 :
3 -2 $22.
2 -4 $28.

?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Conclusion:
x =
y =

Conclusion: 3x + 2y = 22
Conclusion: 2x + 4y = 28

1 -2:
6x + 4y = 44
2x + 4y = 28

Conclusion:
4x = 16
x = 4 ()

Conclusion:
3(4) + 2y = 22
12 + 2y = 22
2y = 10
Question 31
2.50 pts

📝 :
.
45.

?

Explanation:

💡 Detailed Explanation:

No 📝

Conclusion:
x =
y =

Conclusion:
"": x = 2y
" 45": x + y = 45

Conclusion:
2y + y = 45
3y = 45
y = 15 ()

x = 2(15) = 30 ()

Conclusion:
30
Question 32
2.50 pts

📝 :
$1 -$2.
15 .
$24.

?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Conclusion:
x = $1
y = $2

Conclusion:
"15 ": x + y = 15
"$24 ": 1x + 2y = 24

Conclusion:
(x + 2y) - (x + y) = 24 - 15
y = 9

x = 15 - 9 = 6

Conclusion:
6 + 9 = 15 ✓
6×1 + 9×2 = 6 + 18 = 24 ✓

: 6 $1, 9 $2

Question 33
2.50 pts

📝 Word problem:
The perimeter of a rectangle is 36 cm.
The length is 4 cm greater than the width.

Find the dimensions of the rectangle.

Explanation:

💡 Detailed explanation:

Geometry and algebra 📝

Definition:
x = length
y = width

Rectangle perimeter formula:
perimeter = 2(length + width)

Equations:
2(x + y) = 36 → x + y = 18
x = y + 4

Substitute:
(y + 4) + y = 18
2y + 4 = 18
2y = 14
y = 7 (width)

x = 7 + 4 = 11 (length)

Correct answer: Length: 11 cm, Width: 7 cm

Question 34
2.50 pts

📝 :
$40 $60 .
5 $48 .

?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

Conclusion:
x = ($40)
y = ($60)

Conclusion:
: x + y = 5
: 40x + 60y = 48×5 = 240

Conclusion: 40x + 60y = 240
÷20: 2x + 3y = 12

-x + y = 5: x = 5 - y

Conclusion:
2(5-y) + 3y = 12
10 - 2y + 3y = 12
y = 2
x = 3

: 3 , 2

Question 35
2.50 pts

📝 :
200 -2
-200 -4 .

velocity/speed velocity/speed ?

Explanation:

💡 Detailed Explanation:

velocity/speed 📝

Conclusion:
x = velocity/speed
y = velocity/speed

Details:
: 200÷2 = 100
: 200÷4 = 50

Conclusion:
x + y = 100 ( )
x - y = 50 ( )

Conclusion:
2x = 150
x = 75

y = 100 - 75 =
Question 36
2.50 pts

🔢 :
:

0.5x + 0.3y = 2.3
0.2x + 0.4y = 1.8

Explanation:

💡 Detailed Explanation:

Isolate and Substitute! 📝

<
Question 37
2.50 pts

🔢 :
:

x/2 + y/3 = 4
x/4 + y/2 = 3

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

-10:

5x + 3y = 23
2x + 4y = 18

1 -2, 2 -5:
10x + 6y = 46
10x + 20y = 90

Conclusion:
-14y = -44
y = 44/14 ≈ 3.14...

, :
2 × 2.5:
5x + 10y = 45
: 7y = 22...

x=2, y=4:
5(2)+3(4)=10+12=22≠23...
1 × 6:
3x + 2y = 24

2 × 4:
x + 2y = 12

Substitute:
(3x + 2y) - (x + 2y) = 24 - 12
2x = 12
x = 6

Substitute:
6 + 2y = 12
2y = 6
y = 3

Substitute: 6/2
Question 38
2.50 pts

🔢 :
:

2x - y = 7
x + 3y = -8

Explanation:

💡 Detailed Explanation:

Solution 📐

-2x - y = 7:
y = 2x - 7

Step 2:
x + 3(2x - 7) = -8
x + 6x - 21 = -8
7x = -8 + 21
7x = 13

:
1 × 3: 6x - 3y = 21
Conclusion: 7x = 13...

x=1, y=-3:
2(1)-(-3)=2+3=5≠7...

...

: x = 1, y = -3

Question 39
2.50 pts

🔢 :
because ?

y = 5x - 3
3x + 2y = 11

Explanation:

💡 Detailed Explanation:

Smart method selection! 🎯

When a variable is isolated → use substitution!

y = 5x − 3 ← y is already isolated!

So use substitution — it's faster here

✅ Substitute y into the other equation directly
Question 40
2.50 pts

📋 :
system of equations?

Explanation:

💡 Detailed Explanation:

Solution Steps 📝

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