The general form of a linear equation ∈ two variables is ax + by + c = 0, where a, b, and c are constants and a ≠ 0, b ≠ 0.
The equation ax + by + c = 0 can be written as y = (-a/b)x - (c/b), which is the slope-intercept form of the equation, where -a/b is the slope and -c/b is the y-intercept.
The equation of a line passing through the points (x₁, y₁) and (x₂, y₂) can be found using the two-point form: y - y₁ = [(y₂ - y₁)/(x₂ - x₁)](x - x₁).
The slope of a line passing through the points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ - y₁)/(x₂ - x₁).
The equation of a line with slope m and passing through the point (x₁, y₁) is given by y - y₁ = m(x - x₁).

🪤 The 5 Mistakes That Cost Marks

Not checking if the given equation is linear or not before solving it.
Not writing the equation ∈ the standard form ax + by + c = 0.
Forgetting to check for any restrictions on the variables.
Not using the correct method to solve the system of equations.
Not verifying the solution by plugging it back into the original equation.

✏️ 3 Solved PYQs

Question 1: Solve the equation 2x + 3y - 7 = 0 for y. Step 1: Rearrange the equation to isolate y. Step 2: 3y = -2x + 7. Step 3: y = (-2/3)x + 7/3.
Question 2: Find the equation of the line passing through the points (2, 3) and (4, 5). Step 1: Find the slope of the line using the two-point form. Step 2: m = (5 - 3)/(4 - 2) = 1. Step 3: Use the point-slope form to find the equation of the line: y - 3 = 1(x - 2). Step 4: Simplify the equation: y - 3 = x - 2, x - y + 1 = 0.
Question 3: Solve the system of equations x + 2y - 3 = 0 and 2x + 3y + 2 = 0. Step 1: Solve the first equation for x. Step 2: x = -2y + 3. Step 3: Substitute x into the second equation: 2(-2y + 3) + 3y + 2 = 0. Step 4: Simplify and solve for y: -4y + 6 + 3y + 2 = 0, -y + 8 = 0, y = 8. Step 5: Substitute y back into one of the original equations to find x: x + 2(8) - 3 = 0, x + 16 - 3 = 0, x + 13 = 0, x = -13.

🧠 The One Thing Most Students Get Wrong

Most students get the concept of slope and intercept mixed up, and they often have trouble finding the equation of a line given two points or the slope and a point.

👁️ Ayush's Note

Make sure to practice solving systems of linear equations using substitution and elimination methods.
Always check your solution by plugging it back into the original equation.
Use the slope-intercept form of a linear equation to easily identify the slope and y-intercept.
Be careful when finding the equation of a line given two points, as the slope must be calculated correctly.

🔁 Last 5 Minutes Box

Review the formulas for slope, slope-intercept form, and the equation of a line given two points.
Practice solving a few simple linear equations to get a feel for the format.
Make sure to check your work and verify your solutions.
Stay calm and read each question carefully to ensure you understand what is being asked.
Use the process of elimination to narrow down your answer choices if you are unsure.

📝 Practice MCQs

1. What is the equation of the line passing through the points (1, 2) and (3, 4)?

A) x - y - 1 = 0

B) x + y - 3 = 0

C) 2x - y - 1 = 0

D) x - 2y + 1 = 0

Answer: A) x - y - 1 = 0.

2. Solve the equation 3x + 2y - 5 = 0 for y.

A) y = (-3/2)x + 5/2

B) y = (-3/2)x + 2

C) y = (3/2)x - 5/2

D) y = (-2/3)x + 5/3

Answer: A) y = (-3/2)x + 5/2.

3. What is the slope of the line passing through the points (2, 3) and (4, 5)?

A) 1

B) 2

C) 3

D) 4

Answer: A) 1.

4. Solve the system of equations x + 2y - 3 = 0 and 2x + 3y + 2 = 0.

A) x = -13, y = 8

B) x = 13, y = -8

C) x = -8, y = 13

D) x = 8, y = -13

Answer: A) x = -13, y = 8.

5. What is the equation of the line with slope 2 and passing through the point (1, 3)?

A) y - 3 = 2(x - 1)

B) y - 2 = 3(x - 1)

C) y - 1 = 2(x - 3)

D) y - 3 = x - 1

Answer: A) y - 3 = 2(x - 1).

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

📚 Related Topics

Continue your revision with these related guides:

The general form of a linear equation ∈ two variables is ax + by + c = 0, where a, b, and c are constants and a ≠ 0, b ≠ 0.
The equation ax + by + c = 0 can be written as y = (-a/b)x - (c/b), which is the slope-intercept form of the equation, where -a/b is the slope and -c/b is the y-intercept.
The equation of a line passing through the points (x₁, y₁) and (x₂, y₂) can be found using the two-point form: y - y₁ = [(y₂ - y₁)/(x₂ - x₁)](x - x₁).
The slope of a line passing through the points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ - y₁)/(x₂ - x₁).
The equation of a line with slope m and passing through the point (x₁, y₁) is given by y - y₁ = m(x - x₁).