Equations and Inequalities
Section: Algebra | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is an Equation?
An equation contains an equals sign and states that two expressions have the same value. Solving an equation means finding the value (or values) of the unknown that make it true.
- 3x + 5 is an expression - it can be simplified or evaluated, but not "solved".
- 3x + 5 = 20 is an equation - it can be solved to find x.
- Whatever operation you apply to one side of an equation, you must apply to the other side too, to keep it balanced.
Solving Linear Equations
Two-Step Equations
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Question: Solve 3x + 5 = 20.
- Step 1: Subtract 5 from both sides: 3x = 15
- Step 2: Divide both sides by 3: x = 5
- Check: 3(5) + 5 = 20 ✓
- Answer: x = 5
Equations with Brackets
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Question: Solve 2(x + 3) = 16.
- Step 1: Expand the bracket: 2x + 6 = 16
- Step 2: Subtract 6 from both sides: 2x = 10
- Step 3: Divide both sides by 2: x = 5
- Check: 2(5 + 3) = 2 × 8 = 16 ✓
- Answer: x = 5
Unknown on Both Sides
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Question: Solve 5x - 3 = 2x + 9.
- Step 1: Subtract 2x from both sides: 3x - 3 = 9
- Step 2: Add 3 to both sides: 3x = 12
- Step 3: Divide both sides by 3: x = 4
- Check: 5(4) - 3 = 17, and 2(4) + 9 = 17 ✓
- Answer: x = 4
Equations with an Unknown in the Denominator
When the unknown is on the bottom of a fraction, multiply both sides by that unknown first to clear the fraction.
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Question: Solve 12/x = 4.
- Step 1: Multiply both sides by x: 12 = 4x
- Step 2: Divide both sides by 4: x = 3
- Check: 12/3 = 4 ✓
- Answer: x = 3
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Question: Solve 20/(x + 2) = 5.
- Step 1: Multiply both sides by (x + 2): 20 = 5(x + 2)
- Step 2: Expand: 20 = 5x + 10
- Step 3: Subtract 10, then divide by 5: x = 2
- Check: 20/(2 + 2) = 20/4 = 5 ✓
- Answer: x = 2
Writing and Solving Equations from Words
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Question: A number is multiplied by 4, then increased by 7. The result is 39. Find the number.
- Step 1: Let the number be n. Write the equation: 4n + 7 = 39
- Step 2: Subtract 7: 4n = 32
- Step 3: Divide by 4: n = 8
- Answer: n = 8
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Question: I think of a number, subtract 6, then divide by 3. The result is 5. Find the number.
- Step 1: Let the number be n. Write the equation: (n − 6)/3 = 5
- Step 2: Multiply both sides by 3: n − 6 = 15
- Step 3: Add 6: n = 21
- Answer: n = 21
Simultaneous Equations: The Elimination Method
Simultaneous equations are two equations that share the same pair of unknowns. Their solution is the one pair of values that satisfies both equations at once. In the elimination method, add or subtract the equations to remove one unknown.
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Question: Solve x + y = 10 and x − y = 2.
- Step 1: The y-terms have opposite signs, so add the equations: (x + y) + (x − y) = 10 + 2, giving 2x = 12
- Step 2: x = 6
- Step 3: Substitute into x + y = 10: 6 + y = 10, so y = 4
- Check: 6 − 4 = 2 ✓
- Answer: x = 6, y = 4
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Question: Solve 3x + 2y = 16 and x + y = 6.
- Step 1: Multiply the second equation by 2: 2x + 2y = 12
- Step 2: Subtract from the first equation: (3x + 2y) − (2x + 2y) = 16 − 12, giving x = 4
- Step 3: Substitute into x + y = 6: 4 + y = 6, so y = 2
- Check: 3(4) + 2(2) = 12 + 4 = 16 ✓
- Answer: x = 4, y = 2
Simultaneous Equations: The Substitution Method
If one equation is already written as "y = ..." (or can easily be rearranged that way), replace y in the other equation with that expression. This turns two equations in two unknowns into a single equation in just one unknown.
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Question: Solve y = 2x + 3 and y = 5x − 6.
- Step 1: Both equations equal y, so set them equal to each other: 2x + 3 = 5x − 6
- Step 2: Rearrange to solve for x: 3 + 6 = 5x − 2x, so 9 = 3x, giving x = 3
- Step 3: Substitute into y = 2x + 3: y = 2(3) + 3 = 9
- Check: y = 5(3) − 6 = 9 ✓
- Answer: x = 3, y = 9
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Question: Solve 2x + y = 13 and 3x − y = 12.
- Step 1: Rearrange the first equation to make y the subject: y = 13 − 2x
- Step 2: Substitute this into the second equation: 3x − (13 − 2x) = 12
- Step 3: Expand and solve: 3x − 13 + 2x = 12, so 5x = 25, giving x = 5
- Step 4: Substitute into y = 13 − 2x: y = 13 − 2(5) = 3
- Check: 3(5) − 3 = 15 − 3 = 12 ✓
- Answer: x = 5, y = 3
Simultaneous Equations: The Graphical Method
Each linear equation can be drawn as a straight line. The solution to a pair of simultaneous equations is the point where the two lines cross, since that point's coordinates satisfy both equations at once.
The lines x + y = 10 and x − y = 2 intersect at (6, 4) - the same solution found algebraically
- Plot both lines on the same grid, using a table of values for each.
- Read off the coordinates of the point where the lines cross.
- This point (x, y) is the solution to the simultaneous equations.
Writing and Solving Inequalities
An inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign. Inequalities are solved the same way as equations - except that multiplying or dividing both sides by a negative number reverses the inequality sign.
- "A number is at least 5" → n ≥ 5
- "The temperature is less than 10 degrees" → t < 10
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Question: Solve 2x + 3 < 11.
- Step 1: Subtract 3 from both sides: 2x < 8
- Step 2: Divide both sides by 2: x < 4
- Answer: x < 4
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Question: Solve −3x + 6 > 0.
- Step 1: Subtract 6 from both sides: −3x > −6
- Step 2: Divide both sides by −3, and reverse the inequality sign: x < 2
- Answer: x < 2
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