Prep 2 - Mathematics
Unit 2: Linear Functions, Slope & Real Numbers Operations
Linear Relations: Graphing ax + by = c, Finding Ordered Pairs, and Axis Intercepts
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In this lesson, we transition to coordinate geometry and explore 'The Linear Relation Between Two Variables'. A linear relation is an equation of the first degree involving two variables, x and y, and is generally written in standard form as ax + by = c (where a and b are real numbers not both zero). We prove algebraically and geometrically that the graph of any linear relation is always a straight line on the Cartesian plane. To draw this line accurately, we determine at least three distinct ordered pairs (x, y) that satisfy the equation. We also discover how to quickly find the x-intercept (by setting y = 0) and the y-intercept (by setting x = 0), and examine horizontal lines (y = k) and vertical lines (x = k).
1. Standard Form & Ordered Pairs
Structure and plotting of linear relations:
- Standard Form:
ax + by = c, where a, b, c ∈ R, and (a, b) ≠ (0, 0). - Graph Representation: Represented graphically by a straight line extending infinitely in both directions.
- Finding Ordered Pairs: Express one variable in terms of the other (e.g., y = c - ax), choose three convenient values for x (e.g., 0, 1, 2), and calculate the corresponding values of y.
- Three Points Rule: We find at least 3 points so that the third point verifies the accuracy of the line (preventing arithmetic errors).
2. Coordinate Axes & Intercepts
Special straight lines and intersections:
- x-intercept: The point where the line crosses the x-axis. Found by substituting y = 0 → (c/a, 0).
- y-intercept: The point where the line crosses the y-axis. Found by substituting x = 0 → (0, c/b).
- Lines Parallel to Axes:
• Horizontal Line (parallel to x-axis): Equation is y = k. (Equation of x-axis itself is y = 0).
• Vertical Line (parallel to y-axis): Equation is x = k. (Equation of y-axis itself is x = 0). - Line passing through the origin (0, 0): Has the form ax + by = 0 (c = 0).
Graphing & Intercepts Clinic:
Q1: Represent graphically the relation: 2x + y = 4.
Solution:
1. Express y in terms of x: y = 4 - 2x.
2. Choose values: for x = 0 → y = 4; for x = 1 → y = 2; for x = 2 → y = 0.
3. Ordered pairs: (0, 4), (1, 2), (2, 0). Plot points on Cartesian plane and draw the straight line through them.
Solution:
1. Express y in terms of x: y = 4 - 2x.
2. Choose values: for x = 0 → y = 4; for x = 1 → y = 2; for x = 2 → y = 0.
3. Ordered pairs: (0, 4), (1, 2), (2, 0). Plot points on Cartesian plane and draw the straight line through them.
Q2: Find the intersection points of the straight line 3x - 4y = 12 with the coordinate axes.
Solution:
1. With x-axis: put y = 0 → 3x = 12 → x = 4. Point = (4, 0).
2. With y-axis: put x = 0 → -4y = 12 → y = -3. Point = (0, -3).
Solution:
1. With x-axis: put y = 0 → 3x = 12 → x = 4. Point = (4, 0).
2. With y-axis: put x = 0 → -4y = 12 → y = -3. Point = (0, -3).
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linear relation
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العلاقة الخطية بين متغيرين
📖 Def: a first-degree polynomial equation involving two variables whose graph is a straight line
ordered pair
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الزوج المرتب
📖 Def: a pair of numbers (x, y) identifying a point on the Cartesian coordinate plane
Cartesian plane
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المستوى الإحداثي الديكارتي
📖 Def: a plane defined by two perpendicular directed coordinate axes
x-intercept
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نقطة التقاطع مع محور السينات
📖 Def: the coordinate point where a graph intersects the horizontal x-axis (y = 0)
y-intercept
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نقطة التقاطع مع محور الصادات
📖 Def: the coordinate point where a graph intersects the vertical y-axis (x = 0)
origin
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نقطة الأصل (0, 0)
📖 Def: the point of intersection of the coordinate axes with coordinates (0, 0)
horizontal line
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المستقيم الأفقي
📖 Def: a line parallel to the x-axis having equation y = constant
vertical line
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المستقيم الرأسي
📖 Def: a line parallel to the y-axis having equation x = constant
standard form
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الصورة القياسية للمعادلة
📖 Def: the representation ax + by = c with integer coefficients
collinear points
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النقاط على استقامة واحدة
📖 Def: points that lie on the exact same straight line
variable
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المتغير الرياضي
📖 Def: a symbol representing a quantity that can take different numerical values
satisfy the equation
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يحقق المعادلة
📖 Def: a set of values that makes both sides of the equation mathematically equal
Click on an option to check your answer and view the detailed explanation:
1
Which of the following ordered pairs satisfies the linear relation: 2x - y = 5?
2
The graph of any first-degree relation in two variables ax + by = c is represented by:
3
What is the equation of the horizontal x-axis?
4
Find the point of intersection of the line 3x + 2y = 6 with the y-axis:
5
The straight line representing the relation 4x - 5y = 0 passes through which point?
6
What is the equation of the vertical line parallel to the y-axis and passing through (5, -2)?
7
If the ordered pair (k, 2k) satisfies the relation x + y = 12, what is the value of k?
8
How many solutions (ordered pairs) does a single linear equation in two variables have in R × R?
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