Solve each problem. If m varies jointly as x and y, and m=10 when x=2 and y=14, find m when x=21 and y=8.

Verified step by step guidance

1. Understand the concept of joint variation: If \( m \) varies jointly as \( x \) and \( y \), it means \( m \) is directly proportional to the product of \( x \) and \( y \). This can be written as the equation \( m = kxy \), where \( k \) is the constant of proportionality.
2. Use the given values \( m = 10 \), \( x = 2 \), and \( y = 14 \) to find the constant \( k \). Substitute these values into the equation \( m = kxy \) to get \( 10 = k \times 2 \times 14 \).
3. Solve for \( k \) by isolating it on one side of the equation: \( k = \frac{10}{2 \times 14} \).
4. Now that you have \( k \), use it to find \( m \) when \( x = 21 \) and \( y = 8 \). Substitute these values and \( k \) into the equation \( m = kxy \) to get \( m = k \times 21 \times 8 \).
5. Simplify the expression to find the value of \( m \) for the new values of \( x \) and \( y \).

Key Concepts

Joint Variation

Joint variation describes a relationship where a variable depends on the product of two or more other variables. In this case, m varies jointly as x and y means m = kxy, where k is a constant. Understanding this helps set up the equation to find unknown values.

Constant of Variation

The constant of variation (k) is a fixed number that relates the variables in a joint variation equation. It is found by substituting known values of the variables into the equation m = kxy. Once k is determined, it can be used to find m for other values of x and y.

Substitution and Solving Equations

After finding the constant k, substitution involves replacing variables with given values to solve for the unknown. This process requires algebraic manipulation to isolate the desired variable, ensuring accurate calculation of m when x and y change.