Question

The freezing point of water is \(32^{\circ}\) Fahrenheit (F), or \(0^{\circ}\) Celsius (C). The boiling point of water is \(212^{\circ} \mathrm{F}\) or \(100^{\circ} \mathrm{C}\). The curve connecting these two point pairs is a straight line. Use the two-point form of the equation of a straight line to derive an equation connecting degrees Fahrenheit and degrees Celsius. (We use this equation to convert between Fahrenheit and Celsius.)

Short answer

To convert Fahrenheit (F) to Celsius (C), use the equation \(C = \frac{5}{9} \(F - 32\)\).

Step-by-step solution

Identify the Two Points

To use the two-point form, start by identifying the two points that are on the straight line. These are the freezing point of water \((32^\circ F, 0^\circ C)\) and the boiling point of water \((212^\circ F, 100^\circ C)\). These can be considered as points (x_1, y_1) and (x_2, y_2) respectively.

Two-Point Form of the Line Equation

The two-point form of the line equation is \(\frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1}\). Substitute point \((x_1, y_1)\) with the values \((32, 0)\) and point \((x_2, y_2)\) with the values \((212, 100)\).

Substitute the Points

Substitute the points into the equation, to get \(\frac{y-0}{100-0} = \frac{x-32}{212-32}\) or \(\frac{y}{100} = \frac{x-32}{180}\).

Solve for y in terms of x

To find the relationship between degrees Celsius (y) and degrees Fahrenheit (x), solve for y in the equation. Multiply both sides by 100 to get \(y = \frac{100}{180} \(x - 32\)\) or \(y = \frac{5}{9} \(x - 32\)\). This is the equation connecting degrees Fahrenheit to degrees Celsius.

Key concepts

Two-Point Form of a Line

The two-point form of a line is a way to express the equation of a straight line when you are given two points through which the line passes. This is extremely useful in various applications, one of which is converting temperatures from Fahrenheit to Celsius, as these temperature scales can be related linearly.

In mathematical terms, the two-point form of a linear equation is represented as \( \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \), where \( x_1, y_1 \) and \( x_2, y_2 \) are the x and y coordinates of the two points respectively. This form is derived from the basic principle that the ratio of the differences in y-coordinates to the differences in x-coordinates between two points is constant for a straight line, and is essentially the slope of the line.

To make it easier to understand, think of it this way: if you know two locations on a map and you want to find the path connecting them, the two-point form gives you the precise direction for the line that joins these two points, no matter where you're starting from on that line.

Linear Equations

Linear equations form the foundation of various mathematical concepts and are vital in numerous real-life applications, including science, engineering, and finance. A linear equation in two variables, x and y, takes the form of \( ax + by = c \), where a, b, and c are constants. The solutions to these equations represent points on a line in a two-dimensional plane.

The simplicity of linear equations makes them a perfect starting point for exploring relationships between two quantities, in this case, temperature measurements. When expressing one temperature unit in terms of another, such as Fahrenheit to Celsius, the relationship is linear because a change in one unit corresponds to a consistent change in the other unit. This consistent rate of change is captured through the slope of the corresponding line on a graph.

Understanding how to manipulate these equations allows for a deeper comprehension of how changes in one variable affect the other, which is the essence of algebraically finding solutions in many scientific fields.

Temperature Conversion Mathematics

Temperature conversion between Fahrenheit and Celsius is a classic example of linear relationships in mathematics. We can derive a conversion formula by using the two-point form of a line, as temperature conversions are linear.

The standard points used for this conversion are the freezing and boiling points of water. In Fahrenheit and Celsius, these are \(32^\circ F, 0^\circ C)\) and \(212^\circ F, 100^\circ C)\) respectively, which are used in the two-point form equation to derive the conversion formula. After substitution and simplifying, we obtain the linear equation \(y = \frac{5}{9}(x - 32)\), where x is the temperature in Fahrenheit, and y is the temperature in Celsius.

It's important to note that the slope, \(\frac{5}{9}\), indicates how much the temperature in Celsius changes for a one-degree change in Fahrenheit. This equation is accurate and reliable for temperature conversion, and thus, is widely used in both scientific contexts and daily life.