Direct Inverse and Joint Variation Word Problems

TL;DR

Direct, inverse, and joint variation word problems are solved by writing the right equation, solving for the constant k, then plugging in the new values. Direct variation is y = kx, inverse is y = k/x, and joint (direct with two variables) is y = kxz. For example, if y varies directly with x and y = 12 when x = 3, then k = 4, so y = 36 when x = 9. Read on for worked inverse and joint examples.

Transcript

in this video we're going to go over direct inverse and joint variation word problems so the first thing that you need to be able to do is you need to be able to write the equation so we're going to focus on that and then we'll apply that in solving word problems so let's begin if y varies directly with x the equation that you need is y is equal to... Read More

Key Insights

• ❣️ Direct variation equations are of the form y = kx, where y and x are directly proportional.
• 🕡 Inverse variation equations are of the form y = k/x, where y and x are inversely proportional.
• ❣️ Joint variation equations are of the form y = kxr, where y varies jointly with x and r.
• 😉 The constant of variation, k, can be found by using the given values in the equation.
• 🔑 Solving word problems involving variations requires writing the correct equation and using the given values to find the unknown variable.
• 📁 Direct and inverse variations have different effects on the relationship between y and the changing variable.
• 📁 It is important to understand the concept of direct, inverse, and joint variation to correctly solve word problems.

Questions & Answers

Q: How do you solve a joint variation word problem?

Write the equation y = kxz, since "jointly" means direct variation with two variables. Use the first set of values to solve for k, then plug in the new values. For example, if y = 36 when x = 2 and z = 3, then when x = 4 and z = 6, y increases by a factor of 4 (2 × 2) to give y = 144.

Q: How do you set up a direct and inverse variation word problem?

For direct variation (y varies directly with x), write y = kx. For inverse variation (y varies inversely with x), write y = k/x. In direct variation, as x goes up y goes up proportionally; in inverse variation, as x goes up y goes down proportionally.

Q: How do you solve a direct variation problem step by step?

First write y = kx. Then use the given values to solve for k. In the example, x = 3 when y = 12, so dividing 12 by 3 gives k = 4. Finally, plug in the new x value: when x = 9, y = 4 × 9 = 36. Because it is directly proportional, tripling x from 3 to 9 also triples y from 12 to 36.

Q: How do you solve an inverse variation problem?

Write y = k/x, then use the given values to solve for k by cross multiplying. In the example, y = 48 when x = 4, so k = 4 × 48 = 192. To find y when x = 8, compute 192 / 8 = 24. This matches the inverse rule: doubling x from 4 to 8 cuts y in half from 48 to 24.

Q: What is the equation when y varies directly with x but inversely with z?

When y varies directly with x but inversely with z, x goes on top and z goes on the bottom, giving y = kx/z. The variable it varies directly with stays in the numerator, and the variable it varies inversely with moves to the denominator.

Q: How do you write an equation when y varies directly with the square of x?

When y varies directly with the square of x, write y = kx². If instead y varies inversely with the square root of a variable, that root goes in the denominator. The video also covers combined cases like varying directly with the cube of x but inversely with the square of z.

Q: What does "jointly" mean in variation problems?

Whenever you hear the word "jointly," it means direct variation but with two variables. So if y varies jointly with x and z, the equation is y = kxz, and as either x or z goes up, y goes up. You multiply the combined effects of x and z when both change.

Q: What is the difference between direct and inverse variation?

In direct variation (y = kx), y and x are proportional, so as x increases y increases and doubling x doubles y. In inverse variation (y = k/x), as x increases y decreases proportionally, so doubling x cuts y in half. The equations reflect this opposite behavior.

Summary & Key Takeaways

• The video explains the equations for direct and inverse variation: y = kx and y = k/x, respectively.

• It also demonstrates how to write equations for joint variation, where y varies jointly with two variables: y = kxr.

• Several word problems are provided, along with step-by-step solutions to find the value of y for different values of x and z.