Definitions [5]
An equation which contains two variables and the degree of each term containing a variable is one is called a linear equation in two variables.
General Form:
ax + by + c = 0
A set of points in a plane is called a convex set if the line segment joining any two points in the set lies entirely within the set.
If the line segment joining any two points in the set does not completely lie in the set, then it is a non-convex set.
A linear inequality or inequation, which has only one variable, is called a linear inequality or inequation in one variable.
e.g. ax + b < 0, where a ≠ 0, 3x + 4 > 0
An inequality or inequation is said to be linear if each variable occurs in the first degree only and there is no term involving the product of the variables.
e.g. ax + b ≤ 0, ax + by + c > 0, x ≤ 4
Key Points
Steps:
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Multiply one or both equations
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Add or subtract
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Solve
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Substitute
| Case | Result While Solving | Type of Statement | Nature of Pair |
|---|---|---|---|
| Unique solution | You get a specific value of x (or y) | Valid numerical value | Consistent |
| Infinitely many solutions | You get a true statement (e.g. 18 = 18) | Always true | Dependent |
| No solution | You get a false statement (e.g. 0 = 9) | Contradiction | Inconsistent |
Steps:
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Express one variable in terms of the other
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Substitute
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Solve
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Substitute back
| Case | What You Get While Solving | Type of Statement | Nature of Equations |
|---|---|---|---|
| Unique solution | Specific values of x and y | Valid numerical values | Consistent |
| Infinitely many solutions | A true statement like 18 = 18 | Always true | Dependent |
| No solution | A false statement like 0 = 5 | Contradiction | Inconsistent |
