a₁x + b₁y = c₁ and a₂x + b₂y = c₂ are the two linear equations ∈ the form of ax + by = c
To solve these equations, we can use the method of substitution or elimination
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, then the equations are inconsistent and have no solution
If a₁/a₂ = b₁/b₂ = c₁/c₂, then the equations are consistent and have infinitely many solutions
The general solution of the equation ax + by = c is given by x = (c - by)/a and y = (c - ax)/b
The equations can also be represented graphically, and the point of intersection represents the solution
The equations can be solved using cross multiplication: x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁)
If a₁b₂ - a₂b₁ = 0, then the equations are dependent and have infinitely many solutions
If a₁b₂ - a₂b₁ ≠ 0, then the equations are independent and have a unique solution

🪤 The 5 Mistakes That Cost Marks

Not checking if the equations are consistent or inconsistent before solving
Not using the correct method to solve the equations, i.e., substitution or elimination
Not simplifying the equations before solving
Not checking the solution by plugging it back into the original equations
Not representing the equations graphically to visualize the solution

✏️ 3 Solved PYQs

Solve the equations: 2x + 3y = 7 and x - 2y = -3 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = (7 - 3y)/2 Step 3: Substitute x into the second equation: ((7 - 3y)/2) - 2y = -3 Step 4: Simplify and solve for y: (7 - 3y)/2 - 2y = -3 => 7 - 3y - 4y = -6 => -7y = -13 => y = 13/7 Step 5: Substitute y back into one of the original equations to find x: x = (7 - 3 (13/7))/2 => x = (49 - 39)/14 => x = 10/14 => x = 5/7
Solve the equations: x + y = 4 and 2x - 2y = -2 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = 4 - y Step 3: Substitute x into the second equation: 2(4 - y) - 2y = -2 Step 4: Simplify and solve for y: 8 - 2y - 2y = -2 => -4y = -10 => y = 5/2 Step 5: Substitute y back into one of the original equations to find x: x = 4 - 5/2 => x = (8 - 5)/2 => x = 3/2
Solve the equations: 3x - 4y = 7 and 5x + 2y = 17 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = (7 + 4y)/3 Step 3: Substitute x into the second equation: 5((7 + 4y)/3) + 2y = 17 Step 4: Simplify and solve for y: (35 + 20y)/3 + 2y = 17 => 35 + 20y + 6y = 51 => 26y = 16 => y = 8/13 Step 5: Substitute y back into one of the original equations to find x: x = (7 + 4(8/13))/3 => x = (91 + 32)/39 => x = 123/39 => x = 41/13

🧠 The One Thing Most Students Get Wrong

Most students get the concept of dependent and independent equations wrong, and they are not able to identify when the equations have a unique solution or infinitely many solutions
They also struggle with the method of substitution and elimination, and they are not able to apply these methods correctly to solve the equations

👁️ Ayush's Note

To solve a pair of linear equations, first check if the equations are consistent or inconsistent
If the equations are consistent, then use the method of substitution or elimination to solve them
If the equations are inconsistent, then there is no solution
Always check the solution by plugging it back into the original equations
Practice solving different types of equations to become proficient ∈ this topic

🔁 Last 5 Minutes Box

Check if the equations are consistent or inconsistent
Use the correct method to solve the equations
Simplify the equations before solving
Check the solution by plugging it back into the original equations
Practice different types of equations to become proficient ∈ this topic

📝 Practice MCQs

1. What is the solution of the equations: x + y = 4 and 2x - 2y = -2?

A) x = 3/2, y = 5/2

B) x = 5/2, y = 3/2

C) x = 2, y = 2

D) x = 1, y = 3

Answer: A) x = 3/2, y = 5/2

2. What is the solution of the equations: 2x + 3y = 7 and x - 2y = -3?

A) x = 5/7, y = 13/7

B) x = 13/7, y = 5/7

C) x = 2, y = 1

D) x = 1, y = 2

Answer: A) x = 5/7, y = 13/7

3. What is the solution of the equations: 3x - 4y = 7 and 5x + 2y = 17?

A) x = 41/13, y = 8/13

B) x = 8/13, y = 41/13

C) x = 2, y = 1

D) x = 1, y = 2

Answer: A) x = 41/13, y = 8/13

4. What is the condition for the equations to have a unique solution?

A) a₁/a₂ = b₁/b₂ = c₁/c₂

B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

C) a₁b₂ - a₂b₁ = 0

D) a₁b₂ - a₂b₁ ≠ 0

Answer: D) a₁b₂ - a₂b₁ ≠ 0

5. What is the condition for the equations to have infinitely many solutions?

A) a₁/a₂ = b₁/b₂ = c₁/c₂

B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

C) a₁b₂ - a₂b₁ = 0

D) a₁b₂ - a₂b₁ ≠ 0

Answer: A) a₁/a₂ = b₁/b₂ = c₁/c₂

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🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

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Continue your revision with these related guides:

a₁x + b₁y = c₁ and a₂x + b₂y = c₂ are the two linear equations ∈ the form of ax + by = c
To solve these equations, we can use the method of substitution or elimination
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, then the equations are inconsistent and have no solution
If a₁/a₂ = b₁/b₂ = c₁/c₂, then the equations are consistent and have infinitely many solutions
The general solution of the equation ax + by = c is given by x = (c - by)/a and y = (c - ax)/b
The equations can also be represented graphically, and the point of intersection represents the solution
The equations can be solved using cross multiplication: x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁)
If a₁b₂ - a₂b₁ = 0, then the equations are dependent and have infinitely many solutions
If a₁b₂ - a₂b₁ ≠ 0, then the equations are independent and have a unique solution

🪤 The 5 Mistakes That Cost Marks

Not checking if the equations are consistent or inconsistent before solving
Not using the correct method to solve the equations, i.e., substitution or elimination
Not simplifying the equations before solving
Not checking the solution by plugging it back into the original equations
Not representing the equations graphically to visualize the solution

✏️ 3 Solved PYQs

Solve the equations: 2x + 3y = 7 and x - 2y = -3 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = (7 - 3y)/2 Step 3: Substitute x into the second equation: ((7 - 3y)/2) - 2y = -3 Step 4: Simplify and solve for y: (7 - 3y)/2 - 2y = -3 => 7 - 3y - 4y = -6 => -7y = -13 => y = 13/7 Step 5: Substitute y back into one of the original equations to find x: x = (7 - 3 (13/7))/2 => x = (49 - 39)/14 => x = 10/14 => x = 5/7
Solve the equations: x + y = 4 and 2x - 2y = -2 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = 4 - y Step 3: Substitute x into the second equation: 2(4 - y) - 2y = -2 Step 4: Simplify and solve for y: 8 - 2y - 2y = -2 => -4y = -10 => y = 5/2 Step 5: Substitute y back into one of the original equations to find x: x = 4 - 5/2 => x = (8 - 5)/2 => x = 3/2
Solve the equations: 3x - 4y = 7 and 5x + 2y = 17 Step 1: Write down the given equations Step 2: Solve the first equation for x: x = (7 + 4y)/3 Step 3: Substitute x into the second equation: 5((7 + 4y)/3) + 2y = 17 Step 4: Simplify and solve for y: (35 + 20y)/3 + 2y = 17 => 35 + 20y + 6y = 51 => 26y = 16 => y = 8/13 Step 5: Substitute y back into one of the original equations to find x: x = (7 + 4(8/13))/3 => x = (91 + 32)/39 => x = 123/39 => x = 41/13

🧠 The One Thing Most Students Get Wrong

Most students get the concept of dependent and independent equations wrong, and they are not able to identify when the equations have a unique solution or infinitely many solutions
They also struggle with the method of substitution and elimination, and they are not able to apply these methods correctly to solve the equations

👁️ Ayush's Note

To solve a pair of linear equations, first check if the equations are consistent or inconsistent
If the equations are consistent, then use the method of substitution or elimination to solve them
If the equations are inconsistent, then there is no solution
Always check the solution by plugging it back into the original equations
Practice solving different types of equations to become proficient ∈ this topic

🔁 Last 5 Minutes Box

Check if the equations are consistent or inconsistent
Use the correct method to solve the equations
Simplify the equations before solving
Check the solution by plugging it back into the original equations
Practice different types of equations to become proficient ∈ this topic

📝 Practice MCQs

1. What is the solution of the equations: x + y = 4 and 2x - 2y = -2?

A) x = 3/2, y = 5/2

B) x = 5/2, y = 3/2

C) x = 2, y = 2

D) x = 1, y = 3

Answer: A) x = 3/2, y = 5/2

2. What is the solution of the equations: 2x + 3y = 7 and x - 2y = -3?

A) x = 5/7, y = 13/7

B) x = 13/7, y = 5/7

C) x = 2, y = 1

D) x = 1, y = 2

Answer: A) x = 5/7, y = 13/7

3. What is the solution of the equations: 3x - 4y = 7 and 5x + 2y = 17?

A) x = 41/13, y = 8/13

B) x = 8/13, y = 41/13

C) x = 2, y = 1

D) x = 1, y = 2

Answer: A) x = 41/13, y = 8/13

4. What is the condition for the equations to have a unique solution?

A) a₁/a₂ = b₁/b₂ = c₁/c₂

B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

C) a₁b₂ - a₂b₁ = 0

D) a₁b₂ - a₂b₁ ≠ 0

Answer: D) a₁b₂ - a₂b₁ ≠ 0

5. What is the condition for the equations to have infinitely many solutions?

A) a₁/a₂ = b₁/b₂ = c₁/c₂

B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

C) a₁b₂ - a₂b₁ = 0

D) a₁b₂ - a₂b₁ ≠ 0

Answer: A) a₁/a₂ = b₁/b₂ = c₁/c₂

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: