Simultaneous Equations

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टीचर्स के लिए: Simultaneous Equations (Maths [CIE], Core) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स, डायग्राम — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

What are Simultaneous Equations?

Simultaneous equations involve two unknowns (usually x and y) and require two equations to find both values.
The solution is the pair of values that satisfy both equations at the same time.
They are called linear if there are no squared terms (like x² or y²).
Example: 3x + 2y = 11 and 2x – y = 5 have solution x = 3, y = 1.

Solving by Elimination

Elimination removes one variable by making the coefficients of x (or y) the same in both equations.
If the signs in front of the term are the same, subtract the equations.
If the signs are different, add the equations.
After eliminating one variable, solve the resulting equation for the other variable.
Substitute the found value into one original equation to find the second variable.
Always check your solutions in the other original equation.

Solving by Substitution

Substitution involves rearranging one equation to make x or y the subject (e.g., y = 2x – 5).
Substitute this expression into the other equation.
Solve the resulting equation for the remaining variable.
Substitute back to find the other variable.
This method is an alternative to elimination.

Solving Graphically

Plot both equations on the same set of axes (use a table of values or rearrange to y = mx + c).
The point of intersection of the two lines gives the solution (x, y).
Example: 2x – y = 3 and 3x + y = 7 intersect at (2, 1), so x = 2, y = 1.
Graphical method is useful for checking answers but may be less precise.

Solving Graphically

Forming Simultaneous Equations from Word Problems

Introduce two letters (e.g., x and y) to represent the unknowns, stating what each stands for.
Translate the given information into two equations.
Example: '3 apples and 2 bananas cost $1.80' gives 3x + 2y = 180 (in cents).
Solve the equations simultaneously, then answer the question in context (with units).
Sometimes you need to find a further value (e.g., product or total cost) after solving.

Worked Example (Elimination)

Solve: 5x + 2y = 11 and 4x – 3y = 18.
Multiply first by 3: 15x + 6y = 33. Multiply second by 2: 8x – 6y = 36.
Add to eliminate y: 23x = 69 → x = 3.
Substitute x = 3 into 5x + 2y = 11 → 15 + 2y = 11 → 2y = –4 → y = –2.
Check in second: 4(3) – 3(–2) = 12 + 6 = 18 ✓. Solution: x = 3, y = –2.

Worked Example (Forming Equations)

Customer 1: 6 bagels + 12 sausage rolls = £9 → 6b + 12s = 9.
Customer 2: 9 bagels + 10 sausage rolls = £12.30 → 9b + 10s = 12.3.
Eliminate b: multiply first by 3, second by 2 → 18b + 36s = 27 and 18b + 20s = 24.6.
Subtract: 16s = 2.4 → s = 0.15. Substitute: 6b + 1.8 = 9 → b = 1.2.
Cost of 5 bagels and 15 sausage rolls = 5×1.2 + 15×0.15 = 6 + 2.25 = £8.25.

Examiner Tips

Always check your final solutions satisfy both original equations.
Write both solutions together (e.g., x = 3, y = –2) to avoid missing one.
Read the question carefully: sometimes you need to find something else (e.g., product, total cost).
Show all working clearly, especially when multiplying equations.

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 60 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.Solve the simultaneous equations: 3x + 2y = 11 and 2x - y = 5.
Medium
- Ax=3, y=1
- Bx=1, y=3
- Cx=2, y=2.5
- Dx=4, y=-0.5
2.Solve the simultaneous equations: 5x - 2y = 26 and 7x + 6y = 10.
Medium
- Ax=4, y=-3
- Bx=2, y=-8
- Cx=6, y=2
- Dx=-2, y=-18
3.Solve the simultaneous equations: 6x - 3y = 12 and 2x + 3y = 16.
Medium
- Ax=3.5, y=3
- Bx=4, y=4
- Cx=2, y=0
- Dx=1, y=-2
4.Solve the simultaneous equations: 5x + 4y = 10 and 7x - 6y = 43.
Medium
- Ax=4, y=-2.5
- Bx=5, y=-3.75
- Cx=3, y=-1.25
- Dx=2, y=0
5.Esme buys x magazines at $2.45 each and y cards at $3.15 each. She spends $60.55 in total and buys 8 magazines. How many cards does she buy?
Medium
- A13
- B14
- C12
- D15
6.A shop sells pens and notebooks. A pen costs p cents, a notebook costs n cents. On Monday, 5 pens and 4 notebooks cost 450 cents. On Tuesday, 10 pens and 3 notebooks cost 525 cents. Find the cost of a pen.
Medium
- A30 cents
- B45 cents
- C60 cents
- D50 cents
7.Beindu buys 7 apples and 4 bananas for 85 cents, and 3 apples and 8 bananas for 93 cents. Find the cost of an apple (a cents) and a banana (b cents).
Medium
- Aa=7, b=9
- Ba=8, b=7
- Ca=9, b=6
- Da=6, b=10
8.A plant costs p dollars and a bush costs b dollars. Ana buys 2 plants and 4 bushes for $42. Paola buys 7 plants and 9 bushes for $107. Find p and b.
Medium
- Ap=5, b=8
- Bp=8, b=5
- Cp=6, b=7
- Dp=4, b=9

और सवाल

9.Alphonse is x years old and Beatrice is y years old. Three times Alphonse's age equals five times Beatrice's age. Twice Beatrice's age is 4 years more than Alphonse's age. Find Alphonse's age.
Hard
- A20
- B12
- C15
- D10
10.Des thinks of two numbers. Their sum is -6 and their difference is 62. Find the two numbers.
Medium
- A28 and -34
- B34 and -28
- C30 and -36
- D26 and -32
11.Abdallah buys 3 cucumbers and 5 tomatoes for £14.50. Katie buys 5 cucumbers and 4 tomatoes for £15.50. Find the cost of a cucumber.
Medium
- A£1.50
- B£2.00
- C£2.50
- D£1.00
12.What are linear simultaneous equations?
Easy
- AEquations with no squared terms that share the same solutions
- BEquations that have the same slope
- CEquations that have two variables
- DEquations that are both linear
13.Which method involves making the coefficients of one variable the same in both equations?
Easy
- AElimination
- BSubstitution
- CGraphical
- DTrial and error
14.When using elimination, if the signs in front of the term to be eliminated are the same, you should:
Easy
- ASubtract the equations
- BAdd the equations
- CMultiply the equations
- DDivide the equations
15.Solve the simultaneous equations: 3x − 8y = 22 and x + 4y = 4.
Medium
- Ax = 6, y = -0.5
- Bx = -6, y = 0.5
- Cx = 6, y = 0.5
- Dx = -6, y = -0.5
16.Solve the simultaneous equations: 3x − y = 22 and x + 2y = 5.
Medium
- Ax = 7, y = -1
- Bx = 7, y = 1
- Cx = -7, y = 1
- Dx = -7, y = -1
17.Solve the simultaneous equations: 5x − 2y = 44 and 2x + 3y = 10.
Medium
- Ax = 8, y = -2
- Bx = 8, y = 2
- Cx = -8, y = 2
- Dx = -8, y = -2
18.Solve the simultaneous equations: y = 3x + 13 and y = 7x - 3.
Medium
- Ax = 4, y = 25
- Bx = 4, y = 31
- Cx = -4, y = 1
- Dx = 2.5, y = 17.5
19.Solve the simultaneous equations: 8x − 3y = 19 and 4x + 5y = 3.
Medium
- Ax = 2, y = -1
- Bx = 2, y = 1
- Cx = -2, y = 1
- Dx = -2, y = -1
20.Solve the simultaneous equations: x + y = 5 and 2x - y = 4.
Easy
- Ax = 3, y = 2
- Bx = 2, y = 3
- Cx = 1, y = 4
- Dx = 4, y = 1
21.Solve the simultaneous equations: 2x + y = 8 and x - y = 1.
Medium
- Ax = 3, y = 2
- Bx = 2, y = 4
- Cx = 4, y = 0
- Dx = 3, y = -2
22.Solve the simultaneous equations: 3x + 2y = 12 and 5x - 2y = 4.
Medium
- Ax = 2, y = 3
- Bx = 2, y = -3
- Cx = 4, y = 0
- Dx = 1, y = 4.5
23.Solve the simultaneous equations: 4x + 3y = 10 and 2x - 3y = 8.
Medium
- Ax = 3, y = -2/3
- Bx = 3, y = 2/3
- Cx = -3, y = 22/3
- Dx = -3, y = -22/3
24.Solve the simultaneous equations: 2x + 5y = 9 and 3x + 2y = 8.
Medium
- Ax = 2, y = 1
- Bx = 1, y = 2
- Cx = -2, y = 13/5
- Dx = -1, y = 11/5
25.The sum of two numbers is 15 and their difference is 3. Find the numbers.
Easy
- A9 and 6
- B10 and 5
- C8 and 7
- D12 and 3
26.Two angles are complementary (sum 90°). The larger angle is 30° more than twice the smaller. Find the angles.
Medium
- A20° and 70°
- B30° and 60°
- C10° and 80°
- D25° and 65°
27.The sum of two numbers is 19 and their difference is 5. What is the product of the two numbers?
Medium
- A84
- B96
- C68
- D72
28.At a bakery, 6 bagels and 12 sausage rolls cost £9.00. 9 bagels and 10 sausage rolls cost £12.30. Find the cost of 5 bagels and 15 sausage rolls.
Hard
- A£8.25
- B£9.00
- C£7.50
- D£8.75
29.Two angles are complementary (sum 90°). The larger angle is 30° more than twice the smaller. Find the larger angle.
Medium
- A70°
- B60°
- C50°
- D80°
30.Solve the simultaneous equations: y = 2x + 1 and y = 3x - 2.
Medium
- Ax = 3, y = 7
- Bx = 1, y = 3
- Cx = -1, y = -1
- Dx = 2, y = 5
31.Solve the simultaneous equations: 3x + 4y = 10 and 2x - 3y = 1.
Medium
- Ax = 2, y = 1
- Bx = 1, y = 2
- Cx = -2, y = 4
- Dx = 3, y = 0.25
32.Solve the simultaneous equations: x + y = 7 and 2x - y = 5.
Easy
- Ax = 4, y = 3
- Bx = 3, y = 4
- Cx = 2, y = 5
- Dx = 5, y = 2
33.Solve the simultaneous equations: 2x + 3y = 8 and 4x - y = 2.
Medium
- Ax = 1, y = 2
- Bx = 2, y = 1
- Cx = 0.5, y = 2.33
- Dx = 1.5, y = 1.67
34.Linear simultaneous equations involve two unknowns and two equations.
Easy
True or false?
35.When using the elimination method to solve simultaneous equations, if the signs in front of the term you want to eliminate are the same, you should:
Easy
- Aadd the equations
- Bsubtract the equations
- Cmultiply the equations
- Ddivide the equations
36.Which of the following are methods to solve linear simultaneous equations? (Select all that apply)
Easy
- AElimination
- BSubstitution
- CGraphical
- DDifferentiation
- EIntegration
37.Match each description to the correct method for solving simultaneous equations.
Medium
- Make coefficients equal then add/subtract
- Rearrange one equation and substitute into the other
- Plot both equations and find intersection
- Elimination
- Substitution
- Graphical
38.Put the steps for solving by elimination in the correct order for the equations 3x + 2y = 11 and 2x - y = 5.
Medium
- Make the coefficients of x the same (multiply first by 2, second by 3)
- Subtract the equations to eliminate x
- Solve for y
- Substitute y back to find x
39.When solving graphically, the solution to the simultaneous equations is the point where the two lines intersect.
Easy
True or false?
40.Put the steps for solving by elimination in the correct order for the equations 2x + 3y = 7 and 4x - y = 5.
Medium
- Make coefficients of y the same: multiply second equation by 3
- Add the equations: 2x + 12x = 14x, 3y - 3y = 0, 7 + 15 = 22
- Solve for x: 14x = 22, x = 22/14 = 11/7
- Substitute x into original equation to find y
41.Solve the simultaneous equations: 2x + y = 7 and x - y = 2.
Medium
- Ax = 3, y = 1
- Bx = 3, y = -1
- Cx = 1, y = 5
- Dx = 5, y = -3
42.Solve the simultaneous equations: 3x + 2y = 11 and 2x - y = 5. What is the value of x?
Medium
- A3
- B2
- C1
- D4
43.Solve the simultaneous equations: 2x + 3y = 8 and 4x - y = 2. Give x as a number.
Medium
44.Complete the sentence.
Easy
When using elimination, if the signs in front of the term to be eliminated are different, you should ____ the equations.
45.Form two simultaneous equations from this context: The sum of two numbers is 10 and their difference is 2. (Use x and y as the numbers, with x > y.)
Easy
46.The sum of two numbers is 12 and their difference is 4. Find the product of the two numbers.
Medium
47.Complete the sentence about solving simultaneous equations by elimination.
Easy
If the signs in front of the term to be eliminated are the ____, you should subtract the equations. If the signs are ____, you should add the equations.
48.Solve the simultaneous equations: 3x + 2y = 12 and 5x - 2y = 4. Give the value of x.
Medium
49.Solve the simultaneous equations: 2x + 3y = 12 and 4x - y = 10. Give the value of x.
Medium
50.Form two simultaneous equations from this context: The sum of two numbers is 13 and their difference is 3. (Use x and y as the numbers, with x > y.)
Easy
____ and ____
51.Solve the simultaneous equations: y = 2x + 5 and y = 3x - 1. Write your answer as 'x = ..., y = ...'.
Medium
52.Solve the simultaneous equations: 3x + 4y = 10 and 2x - 3y = 1. Give the value of x.
Medium
53.Explain how to solve the simultaneous equations 3x + 2y = 11 and 2x - y = 5 using the elimination method.
Hard
54.Solve the simultaneous equations: 2x + 3y = 8 and 4x - y = 2. Give x as a number.
Medium
- A1
- B3
- C2
- D0
55.The sum of two numbers is 12 and their difference is 4. Find the product of the two numbers.
Medium
- A35
- B38
- C32
- D29
56.Solve the simultaneous equations: 3x + 2y = 12 and 5x - 2y = 4. Give the value of x.
Medium
- A2
- B1
- C4
- D3
57.Solve the simultaneous equations: 2x + 3y = 12 and 4x - y = 10. Give the value of x.
Medium
- A2
- B3
- C5
- D4
58.Solve the simultaneous equations: 3x + 4y = 10 and 2x - 3y = 1. Give the value of x.
Medium
- A3
- B2
- C1
- D4
59.The sum of two numbers is 12 and their difference is 2. Find the numbers.
Medium
- A7 and 5
- B8 and 4
- C6 and 6
- D9 and 3
60.The graph shows the lines y = x + 1 and y = -x + 5. What is the solution to the simultaneous equations?
Medium
- Ax = 2, y = 3
- Bx = 1, y = 2
- Cx = 3, y = 4
- Dx = 0, y = 1

रिवीज़न नोट्स

## Simultaneous Equations - Quick Revision ### Solving by Elimination 1. Make coefficients of one variable equal (multiply equations if needed). 2. If signs are **same**, subtract; if **different**, add. 3. Solve for the remaining variable. 4. Substitute back to find the other variable. 5. Check in both original equations. ### Solving by Substitution 1. Rearrange one equation to y = ... or x = ... 2. Substitute into the other equation. 3. Solve for the variable. 4. Substitute back to find the other. ### Solving Graphically - Plot both lines; the intersection point gives (x, y). ### Forming Equations from Word Problems - Define variables (e.g., x = price of an apple, y = price of a banana). - Write two equations from given information. - Solve simultaneously. - Answer in context with units. ### Key Points - Always show working. - Write final answer as x = ..., y = ... - Check solutions satisfy both equations. ### Example (Elimination) Solve: 5x + 2y = 11 (1) 4x - 3y = 18 (2) Multiply (1) by 3, (2) by 2: 15x + 6y = 33 8x - 6y = 36 Add: 23x = 69 → x = 3 Substitute into (1): 5(3)+2y=11 → 2y=-4 → y=-2 Answer: x=3, y=-2

Simultaneous equations: Two equations with two unknowns (x and y) that are both true at the same time.
Elimination method: A method to solve simultaneous equations by adding or subtracting the equations to remove one variable.
Substitution method: A method to solve simultaneous equations by rearranging one equation and substituting it into the other.
Coefficient: The number in front of a variable (e.g., in 3x, the coefficient is 3).

फ्लैशकार्ड

What are linear simultaneous equations?

Two equations with two unknowns (x and y) that are solved together because the same values satisfy both equations at the same time.

Solve: 3x − 8y = 22 and x + 4y = 4

x = 6, y = -0.5 (or y = -1/2)

Solve: 3x − y = 22 and x + 2y = 5

x = 7, y = -1

Solve: 5x − 2y = 26 and 7x + 6y = 10

x = 4, y = -3

Solve: 6x − 3y = 12 and 2x + 3y = 16

x = 3.5, y = 3

Solve: 5x − 2y = 44 and 2x + 3y = 10

x = 8, y = -2

Solve: 5x + 4y = 10 and 7x − 6y = 43

x = 4, y = -2.5

Find the intersection point of y = 3x + 13 and y = 7x − 3.

x = 4, y = 25

Solve: 8x − 3y = 19 and 4x + 5y = 3

x = 2, y = -1

Solve: 5x + 2y = 11 and 4x − 3y = 18

x = 3, y = -2

What is the elimination method for solving simultaneous equations?

Make the coefficients of one variable the same in both equations, then add or subtract to eliminate that variable and solve for the other.

When eliminating, when do you add equations?

When the signs in front of the term you want to eliminate are different.

When eliminating, when do you subtract equations?

When the signs in front of the term you want to eliminate are the same.

How do you solve simultaneous equations by substitution?

Rearrange one equation to make x or y the subject, then substitute that expression into the other equation.

How do you solve simultaneous equations graphically?

Plot both lines on the same axes; the intersection point gives the solution (x, y).

Esme buys x magazines at $2.45 each and y cards at $3.15 each. Total cost $60.55, she buys 8 magazines. How many cards?

13 cards (2.45×8 + 3.15y = 60.55 → y = 13)

5 pens and 4 notebooks cost 450 cents. 10 pens and 3 notebooks cost 525 cents. Find cost of pen and notebook.

Pen = 30 cents, Notebook = 75 cents

7 apples and 4 bananas cost 85 cents. 3 apples and 8 bananas cost 93 cents. Find cost of apple and banana.

Apple = 7 cents, Banana = 9 cents

6 adult tickets and 2 child tickets cost $124. 3 adult and 5 child cost $100. Find adult and child ticket prices.

Adult = $17.50, Child = $9.50

2 plants and 4 bushes cost $42. 7 plants and 9 bushes cost $107. Find p and b.

p = 5, b = 8

Three times Alphonse's age equals five times Beatrice's age. Twice Beatrice's age is 4 years more than Alphonse's age. Find their ages.

Alphonse = 20, Beatrice = 12

Two numbers sum to -6 and differ by 62. Find the numbers.

28 and -34

Minibus hire: 3 hours, 10 passengers costs $61; 5 hours, 8 passengers costs $80. Find h and p.

h = 12, p = 2.5

3 cucumbers and 5 tomatoes cost £14.50. 5 cucumbers and 4 tomatoes cost £15.50. Find cost of each.

Cucumber = £1.50, Tomato = £2.00

6 bagels and 12 sausage rolls cost £9. 9 bagels and 10 sausage rolls cost £12.30. Find cost of 5 bagels and 15 sausage rolls.

£8.25

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