IB Class 9 Linear Equations Worksheet PDF helps students understand equation solving methods clearly. The worksheet includes structured exercises and word problems. Moreover, learners practice balancing equations and simplifying expressions. This strengthens algebra fundamentals effectively.
Improve Accuracy and Logical Reasoning
IB Class 9 Linear Equations Worksheet PDF supports systematic exam preparation. Therefore, students gain confidence while solving one-variable equations. Additionally, regular practice improves speed and precision. As a result, learners perform better in assessments. Step-by-step solutions ensure deeper conceptual understanding.
Linear Equations and Inequalities Practice
Students can use IB Maths Inequalities Questions with Answers PDF for extended revision. Moreover, Linear Equations and Inequalities IB Practice Questions provide structured exercises. Therefore, consistent practice strengthens algebra skills and improves overall exam readiness.
Linear Equations and Inequalities
Linear equations and inequalities are the foundation of algebra. They help us model real-world situations where relationships between variables are constant. This chapter covers solving equations, understanding inequalities, and representing solutions on number lines and graphs.
1. Linear Equations in One Variable
A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a ≠ 0. The variable has an exponent of 1.
Solving Simple Linear Equations
The goal is to isolate the variable on one side of the equation using inverse operations.
Example 1: Solve 3x + 5 = 20
3x + 5 = 20
3x = 20 – 5 = 15
x = 15 ÷ 3 = 5
Example 2: Solve 7 – 2x = 3x – 8
7 – 2x = 3x – 8
7 + 8 = 3x + 2x (collect x terms on one side)
15 = 5x
x = 3
Check: Substitute x=3: 7-2(3)=1 and 3(3)-8=1 ✓
Equations with Fractions and Decimals
Clear fractions by multiplying both sides by the lowest common denominator (LCD).
A linear equation in two variables (x and y) can be written as ax + by + c = 0 or y = mx + c. The graph of such an equation is a straight line.
Slope-Intercept Form: y = mx + c
m = slope (gradient) = rise/run = (change in y)/(change in x)
c = y-intercept (where the line crosses the y-axis)
Example: Identify slope and y-intercept of y = -2x + 5
Slope m = -2, y-intercept c = 5
The line goes down 2 units for every 1 unit right, crossing the y-axis at (0,5).
Finding the Equation of a Line
Given two points, we can find the slope and then the equation.
Example: Find the equation of the line passing through (1,3) and (4,9).
Slope m = (9-3)/(4-1) = 6/3 = 2
Using y = mx + c: 3 = 2(1) + c → c = 1
Equation: y = 2x + 1
3. Solving Simultaneous Linear Equations
When we have two linear equations with the same two variables, we look for the point (x, y) that satisfies both equations.
Method 1: Substitution
Solve one equation for one variable, then substitute into the other.
Example: Solve y = 2x + 1 and 3x + 2y = 16
Substitute y: 3x + 2(2x + 1) = 16
3x + 4x + 2 = 16 → 7x = 14 → x = 2
Then y = 2(2) + 1 = 5
Solution: (2, 5)
Method 2: Elimination
Add or subtract equations to eliminate one variable.
Example: Solve 2x + 3y = 12 and 4x – 3y = 6
Add the equations: (2x+3y)+(4x-3y)=12+6
6x = 18 → x = 3
Substitute: 2(3)+3y=12 → 6+3y=12 → 3y=6 → y=2
Solution: (3, 2)
Graphical Interpretation
The solution is the point where the two lines intersect.
One solution: Lines intersect at one point (different slopes).
No solution: Lines are parallel (same slope, different intercepts).
Infinite solutions: Lines are coincident (same line).
4. Linear Inequalities in One Variable
An inequality compares two expressions using symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal).
Solving Linear Inequalities
Solve similarly to equations, but with one crucial difference:
Important: When multiplying or dividing both sides by a negative number, reverse the inequality sign.
Example 1: Solve 3x – 7 ≤ 8
3x ≤ 15
x ≤ 5
On a number line: closed circle at 5, shade left.
Example 2 (Negative multiplication): Solve 5 – 2x > 9
-2x > 4
Divide by -2 (reverse sign): x < -2
On a number line: open circle at -2, shade left.
Compound Inequalities
Two inequalities joined by “and” or “or”.
Example (And): -3 < 2x + 1 ≤ 7
Subtract 1: -4 < 2x ≤ 6
Divide by 2: -2 < x ≤ 3
This means x > -2 and x ≤ 3. On number line: open at -2, closed at 3, shade between.
5. Linear Inequalities in Two Variables
An inequality like y > 2x – 1 represents a region of the coordinate plane, not just a line.
Graphing Linear Inequalities
Graph the boundary line (treat as equation y = mx + c).
Use a dashed line for < or > (points on line not included).
Use a solid line for ≤ or ≥ (points on line included).
Test a point (like (0,0)) to decide which side to shade.
Shade the region that satisfies the inequality.
Example: Graph y ≤ 2x – 1
Boundary: y = 2x – 1 (solid line)
Test (0,0): 0 ≤ 2(0)-1? 0 ≤ -1? False
Shade the side opposite to (0,0).
6. Real-World Applications
Solved Examples
Example 1 (Budgeting): A student has $50 to spend on books (b) and pens (p). Books cost $10 each, pens $2 each. Write an inequality for the number of books and pens they can buy.
10b + 2p ≤ 50
Simplify: 5b + p ≤ 25. This is a linear inequality in two variables.
Example 2 (Temperature conversion): The formula to convert Celsius (C) to Fahrenheit (F) is F = 9/5C + 32. Find the temperature range in Celsius if Fahrenheit is between 50° and 86°.
50 < 9/5C + 32 < 86
Subtract 32: 18 < 9/5C < 54
Multiply by 5/9: 10 < C < 30
So, between 10°C and 30°C.
Example 3 (Break-even point): A company’s revenue R = 25x and cost C = 15x + 200, where x is units sold. Find the number of units needed to make a profit (R > C).
25x > 15x + 200
10x > 200
x > 20
They need to sell more than 20 units to make a profit.
7. Common Pitfalls
Incorrect: Forgetting to reverse inequality when multiplying/dividing by a negative. ✓ Correct: -2x < 6 → x > -3 (sign flips).
Incorrect: Using a solid line for a strict inequality (< or >). ✓ Correct: Use dashed line for strict, solid for ≤ or ≥.
Incorrect: Shading the wrong region when graphing inequalities. ✓ Correct: Always test a point not on the line.
Incorrect: Confusing “and” vs “or” in compound inequalities. ✓ Correct: “And” means intersection (both true). “Or” means union (either true).
8. Practice Questions
Solve 5(x – 3) = 2(x + 6).
Solve the inequality: 7 – 3x ≥ 13.
Find the equation of the line through (2,5) and (4,11).
Solve the system: 2x + y = 10 and x – y = 2.
Graph the inequality y > -x + 2.
A taxi charges $3 base fare plus $2 per km. Write an inequality for the maximum distance you can travel with $20.
Answers: 1) x=9 2) x ≤ -2 3) y = 3x – 1 4) (4,2) 5) Dashed line through (0,2) and (2,0), shade above 6) 3 + 2d ≤ 20 → d ≤ 8.5 km
Grade 9 Mathematics
Expansion and Factorisation: Word Problems (Level 1)
Question 1: The length of a rectangular garden is (3x + 4) meters and its width is (2x – 1) meters. Find an expression for the area of the garden in expanded form. If x = 5, calculate the actual area.
Answer: Area = 6x2 + 5x – 4 m2; When x = 5, area = 171 m2
Question 2: A number when multiplied by (x + 3) gives (x2 + 7x + 12). Find the number.
Answer: (x + 4)
Question 3: The product of two consecutive integers is 56 more than the square of the smaller integer. If the smaller integer is n, form an equation and find the integers.
Answer: n(n+1) = n2 + 56; Integers are 56 and 57
Question 4: A square piece of cardboard has side length (2x + 3) cm. A small square of side x cm is cut from each corner. Write an expression for the remaining area in factorised form.
Answer: 3(4x + 3) cm2
Question 5: The sum of two numbers is 15 and their product is 56. If one number is a and the other is (15 – a), form a quadratic equation and find the numbers.
Answer: a2 – 15a + 56 = 0; Numbers are 7 and 8
Question 6: A rectangular swimming pool has length (x + 6) meters and width (x – 2) meters. A path of uniform width x meters surrounds the pool. Find an expression for the total area including the path.
Answer: 9x2 + 12x – 12 m2
Question 7: The difference between two numbers is 4 and the sum of their squares is 136. If the smaller number is y, form an equation and find the numbers.
Answer: (y+4)2 + y2 = 136; Numbers are 6 and 10
Question 8: A rectangular field has an area of (x2 + 5x – 24) m2. If the length is (x + 8) meters, find the width in terms of x.
Answer: Width = (x – 3) meters
Question 9: The product of (x + a) and (x + b) is x2 + 11x + 30. Find the values of a and b where a and b are positive integers with a > b.
Answer: a = 6, b = 5
Question 10: A number when added to its reciprocal gives 41/20. If the number is p, form a quadratic equation and find the number.
Answer: 20p2 – 41p + 20 = 0; Number is 5/4 or 4/5
Question 11: The length of a rectangle is 3 cm more than twice its width. If the area of the rectangle is 65 cm2, find the dimensions.
Answer: Width = 5 cm, Length = 13 cm
Question 13: The sum of the squares of two consecutive odd numbers is 202. Find the numbers.
Answer: 9 and 11
Question 14: A rectangular box has a volume of (x3 + 3x2 – 4x – 12) cm3. If the height is (x – 2) and the width is (x + 2), find the length.
Answer: Length = (x + 3) cm
Question 15: A garden measuring 12m by 16m has a pathway of width x meters around its perimeter. The total area is 285 m2. Find x.
Answer: 4x2 + 56x – 93 = 0; x = 1.5 m
