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Section 1: The Concept and Algebraic Formula of Direct Variation
A variable (y) is said to vary directly as variable (x), written mathematically as [y proportional_to x], if y changes in response to x such that their quotient remains a non-zero constant 'm': [y / x = m], or algebraically: [y = m * x], where 'm' is the (Constant of Direct Variation) and m != 0. This means that if x doubles, y doubles; if x triples, y triples; and if x decreases by half, y decreases by the exact same proportion.
Quick Concept Check
Question: What is the algebraic formula for direct variation y proportional_to x? What is the condition on m?
Key Answer: The formula is y = m * x or y / x = m, where m is a non-zero real constant (m != 0).
Section 2: The Proportional Ratio Property
If y varies directly as x, and y takes the values y1 and y2 when x takes x1 and x2 respectively, then the proportional relationship holds: [y1 / y2 = x1 / x2] or [y1 / x1 = y2 / x2 = m]. This two-state formula provides a direct method for finding an unknown value without having to explicitly determine the equation of variation first.
Quick Concept Check
Question: State the ratio formula for direct variation between two states (x1, y1) and (x2, y2).
Key Answer: y1 / y2 = x1 / x2 (or y1 / x1 = y2 / x2).
Section 3: Graphical Representation of Direct Variation
Direct variation [y = m * x] is represented graphically on the Cartesian plane by a (Straight Line Passing Through the Origin (0, 0)). The constant 'm' represents the slope of the line:
1. If m > 0 (positive): The line ascends and makes an acute angle with the positive direction of the x-axis.
2. If m < 0 (negative): The line descends and makes an obtuse angle.
Crucial Geometric Rule: Any straight line that does not pass through the origin (such as y = 2x + 3) does NOT represent direct variation, because the ratio y/x is not constant.
Quick Concept Check
Question: What is the mandatory geometric condition for a line graph to represent direct variation?
Key Answer: The straight line must pass directly through the Origin (0, 0).
Section 4: Physics and Real-World Applications
Direct variation governs many fundamental physical laws:
1. (Distance and Time): An object moving at a uniform speed (v) covers a distance (d) directly proportional to time (t): [d = v * t => d proportional_to t].
2. (Ohm's Law in Electricity): Potential difference (V) varies directly as electric current intensity (I) at constant resistance (R): [V = R * I => V proportional_to I].
3. (Circumference of a Circle): The circumference (C) varies directly as the radius (r): [C = 2 * pi * r => C proportional_to r].
Quick Concept Check
Question: Give two real-world physical examples of direct variation and identify the constant.
Key Answer: Distance and time at constant speed (d = vt, constant is v), and Ohm's Law (V = IR, constant is R).
Mathematical Vocabulary & Glossary
Master essential mathematical terms with native pronunciation and examples (12 Terms)
Definition
Direct Variation
A relationship between variables where their quotient y/x is constant
Example: y varies directly as x (y proportional_to x).
Definition
Proportionality Sign
The mathematical symbol proportional_to denoting variation
Example: y proportional_to x.
Definition
Variation Constant (m)
The non-zero real ratio m = y / x
Example: In y = 4x, the variation constant is 4.
Definition
Origin (0, 0)
The intersection point through which every direct linear graph must pass
Example: Direct variation lines cross (0, 0).
Definition
Independent Variable
Variable x that varies freely
Example: Time is an independent variable.
Definition
Dependent Variable
Variable y whose value depends directly on x
Example: Distance depends on time.
Definition
Mathematical Slope
The steepness of the line equal to constant m
Example: Slope = m = y/x.
Definition
Ohm's Law
A physical direct variation law V = I * R
Example: Voltage varies directly as current.
Definition
Direct Proportionality
A synonym for direct linear variation
Example: Direct proportionality between force and acceleration.
Definition
Two-State Formula
y1 / y2 = x1 / x2 used to find unknown values
Example: Apply two-state ratio to solve for y2.
Definition
Uniform Speed
Constant velocity serving as the variation scalar in d = vt
Example: Uniform speed produces direct variation.
Definition
Collinear
Points that lie together on a single straight line through the origin
Example: Data points are collinear.
Deep Conceptual Analysis & Models
Comprehensive breakdown of core concepts, charts, and mathematical relationships
Golden Rules of Direct Variation (y proportional_to x)
• Algebraic Formula: y = mx (m is real constant != 0).
• Constant Formula: m = y / x.
• Two-State Ratio: y1 / y2 = x1 / x2.
• Line Graph: Straight line passing through the Origin (0, 0).
• Slope: Slope of the line equals the variation constant m.
• Constant Formula: m = y / x.
• Two-State Ratio: y1 / y2 = x1 / x2.
• Line Graph: Straight line passing through the Origin (0, 0).
• Slope: Slope of the line equals the variation constant m.
Comparative Reference: Recognizing Direct Variation in Equations & Tables
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| Format | Direct Variation Criterion | Valid Example (Direct) | Invalid Example (Not Direct) |
|---|---|---|---|
| Numerical Table | y/x is constant across all pairs | (2, 6), (3, 9) => y/x = 3 | (2, 5), (4, 7) => y/x varies |
| Algebraic Equation | y = mx (no constant term) | y = 5x , 2y - 7x = 0 | y = 3x + 4 (contains +4) |
| Line Graph | Passes through the origin (0, 0) | Line intersecting (0, 0) | Line intersecting y-axis at (0, 2) |
Algebra Lab: Finding the Direct Variation Equation and Unknowns
Standard two-step ministerial method for solving direct variation problems:
1. Determine the Constant m
Since y proportional_to x => y = mx => m = y / x (using initial values).
Example: y = 14 when x = 2 => m = 14 / 2 = 7.
2. Formulate the Equation and Substitute
Write the equation y = 7x, then substitute the target x-value.
Example: When x = 5 => y = 7(5) = 35.
3. Verification
Verify quotient constancy: 14/2 = 7 and 35/5 = 7 (Verified).
Example: Quotient equality guarantees full marks.
Guided Practice Challenge
Problem:
Mastery Assessment Bank (10 Questions)
Select the correct answer for each problem. Immediate pedagogical feedback will be provided.
Score: 0 / 10
1
If y proportional_to x and y = 6 when x = 2, then the relation between x and y is:
2
Which of the following relations represents a direct variation between x and y?
3
If y proportional_to x and y = 15 when x = 3, then the value of y when x = 5 is:
4
The graph of the direct variation y = mx is a straight line that:
5
If (x1, y1) and (x2, y2) are ordered pairs in direct variation, then:
6
If y varies directly as x and y = 2 when x = 8, then the variation constant m is:
7
Which of the following tables represents a direct variation between x and y?
8
If d varies directly as t (d = vt), and a car travels 180 km in 3 hours, then the distance traveled in 5 hours at the same speed is:
9
If y - 3x = 0, then the type of variation between x and y is:
10
If y^2 proportional_to x^2 for positive values, then: