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Linear equations are mathematical expressions that represent a straight line when plotted on a graph. The general form of a linear equation in two variables is given by: \[ y = mx + c \] Here, \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( c \) is the y-intercept (the point where the line crosses the y-axis). Linear equations play a crucial role in various real-world applications, from physics to economics.
In the given exercise, the linear equation describes the relationship between an adult female's height \( H \) and the length of her femur \( x \). Specifically, \( H(x) = 2.47x + 54.10 \) formats the equation to calculate height as a function of femur length.

Height estimation using a linear equation involves substituting a specific value for the femur length into the formula and calculating the resulting height. In this case, the formula provided is: \[ H(x) = 2.47x + 54.10 \] For part (a) of the exercise, we need to find the height for a femur length of 46.8 centimeters. By substituting 46.8 into \( x \): \[ H(46.8) = 2.47 \times 46.8 + 54.10 \] After multiplying and adding the constants, we find that the approximate height is 169.7 centimeters. This method is very useful, especially in forensic science, anthropology, and medical fields where only partial skeletal remains are available.

The substitution method involves replacing a variable with its given value in an equation to solve for another variable. This method is widely used when working with linear equations.
In part (a) of the exercise, we used the substitution method to estimate height from femur length. Conversely, in part (b), we need to estimate femur length for a given height using the equation: \[ 152.4 = 2.47x + 54.10 \] Step by step, we solve this equation:

• Subtract 54.10 from both sides: \[ 152.4 - 54.10 = 2.47x \]
• Simplify to find: \[ 98.3 = 2.47x \]
• Divide both sides by 2.47: \[ x = \frac{98.3}{2.47} \]

Perform the division to find \( x \) approximately 39.8 centimeters. This approach helps to find the missing variable when the other variable and the linear relationship are known.

Calculating the femur length from height involves rearranging the linear formula and solving for \( x \). Given a height, we need to solve for the femur length using: \[ H(x) = 2.47x + 54.10 \] For example, in part (b) of the exercise, the height is 152.4 centimeters. By setting up the equation: \[ 152.4 = 2.47x + 54.10 \] We rearrange and solve for \( x \) as follows:

• Subtract 54.10 from both sides: \[ 152.4 - 54.1 = 2.47x \]
• Simplify to get: \[ 98.3 = 2.47x \]
• Divide by 2.47: \[ x = \frac{98.3}{2.47} \ x \ \text{approximately} \ 39.8 \ \text{centimeters} \]

Thus, the femur length is found to be approximately 39.8 centimeters. This method is significant in fields like anthropology and forensic science where determining body measurements from partial data is essential.