How to solve direct variation

Summary

This video explains the concept of variation, focusing on direct variation, where one quantity changes in proportion to another. Using real-life examples like the cost of exercise books, it demonstrates how to express relationships mathematically as y = kx or y = kx², find the constant of proportionality, and calculate unknown values. The video also shows how to complete tables using direct variation formulas and solve more complex problems involving percentage increases when variables change by a given factor. Step-by-step methods are provided to help viewers understand and apply these concepts effectively, preparing them for similar questions in exams.

Takeaways

• 😀 Variation describes how one quantity changes in relation to another.
• 😀 Direct variation occurs when an increase in one quantity causes a proportional increase in another.
• 😀 The general formula for direct variation is y = kx, where k is the constant of proportionality.
• 😀 To find the constant of proportionality, substitute known values of x and y into the formula.
• 😀 Example: If y = 9 when x = 3, then k = 3 and the relation becomes y = 3x.
• 😀 Direct variation can also involve powers of x, such as y varying directly as x², giving the formula y = kx².
• 😀 To complete a table of values for y, substitute each x value into the direct variation formula.
• 😀 Percentage increase in quantities under direct variation can be calculated by substituting the increased value and comparing with the original value.
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• 😀 For example, if a varies directly as r² and r increases by 20%, the new value of a is 1.44 times the original, representing a 44% increase.
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• 😀 Similarly, if volume V varies directly as r³ and r increases by 20%, V increases by 72.8%, illustrating the impact of higher powers on percentage change.
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• 😀 Using the concept of direct variation allows solving practical problems in mathematics and related fields efficiently.

Q & A

What is the definition of variation as explained in the video?

Variation is defined as how things are tied together or how a change in one quantity affects a change in another quantity.

What is direct variation?

Direct variation occurs when one quantity increases or decreases proportionally with another. In other words, as one quantity increases, the other also increases in direct proportion.

How is direct variation represented mathematically?

Direct variation is represented by the formula y = kx, where k is the constant of proportionality. The symbol '∝' for proportionality is replaced by k to form the equation.

How do you find the constant of proportionality (k) in a direct variation problem?

To find k, substitute the given values of x and y into the formula y = kx (or y = kx² for quadratic variation) and solve for k.

In the example where y = 9 when x = 3, what is the value of k and how do you find y when x = 4?

Using y = kx, we have 9 = 3k, so k = 3. Substituting x = 4 gives y = 3*4 = 12.

How do you handle direct variation when y varies as the square of x?

For quadratic direct variation, the formula is y = kx². You find k by substituting a known pair of values for x and y, then use the formula to calculate other y values for different x values.

How do you complete a table for y = kx² once k is known?

Substitute each x value from the table into the formula y = kx² and calculate y. For example, if k = 3 and x = 3, then y = 3*(3²) = 27.

How can direct variation be applied to real-world problems involving percentage increase?

When a quantity varies directly as another (e.g., area or volume with radius), you can calculate the new value after an increase in the independent variable and then find the percentage increase using: (new value - original value)/original value * 100%.

In the example where A ∝ r² and r increases by 20%, what is the percentage increase in A?

New radius r' = 1.2r. Then A' = k*(1.2r)² = 1.44k*r² = 1.44A. The percentage increase = (1.44 - 1)*100% = 44%.

In the example where V ∝ r³ and r increases by 20%, what is the percentage increase in V?

New radius r' = 1.2r. Then V' = k*(1.2r)³ = 1.728V. The percentage increase = (1.728 - 1)*100% = 72.8%.

Why is it important to replace the proportionality sign with a constant in direct variation problems?

Replacing the proportionality sign with a constant k allows us to write an actual equation that can be used for calculations, making it possible to find unknown values systematically.

What is the general procedure to solve any direct variation problem according to the video?

Step 1: Identify the type of variation. Step 2: Write the proportional relationship with constant k. Step 3: Solve for k using given values. Step 4: Substitute k to find unknowns. Step 5: Apply to tables or percentage increase problems if required.