Problem 28 - Algebra 2

Question

Ice forms at a temperature of \(0^{\circ} \mathrm{C},\) which corresponds to a temperature of \(32^{\circ} \mathrm{F}\). A temperature of \(100^{\circ} \mathrm{C}\) corresponds to a temperature of \(212^{\circ} \mathrm{F} .\) Write and graph the linear equation that gives the number \(y\) of degrees Fahrenheit in terms of the number \(x\) of degrees Celsius.

Step-by-Step Solution

Answer

The linear equation is \(y = 1.8x + 32\).

Step 1: Identify the given points

We are given two temperature points to use for converting Celsius to Fahrenheit: \( (0, 32) \) and \( (100, 212) \), where the first number in each pair is the temperature in degrees Celsius and the second is in degrees Fahrenheit.

Step 2: Find the slope of the line

The slope \( m \) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Using our points \((0, 32)\) and \((100, 212)\), we find:\[m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8\]

Step 3: Write the equation in point-slope form

Using the point-slope form of a line, \(y - y_1 = m(x - x_1)\), and the point \((0, 32)\), our equation is:\[y - 32 = 1.8(x - 0)\]

Step 4: Convert to slope-intercept form

Rearrange the equation from the previous step to get the slope-intercept form \(y = mx + b\):\[y = 1.8x + 32\]

Step 5: Plot the equation

To graph the equation \(y = 1.8x + 32\), plot the two given points \((0, 32)\) and \((100, 212)\) on a coordinate plane and draw a line through them. The line extends infinitely in both directions but represents the relationship between Celsius and Fahrenheit.

Key Concepts

Slope

In mathematics, the slope of a line is a measure of its steepness. Think of it as how much the line rises (or falls) as you move along it from left to right. The slope is usually denoted by the letter \(m\). It is calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. The difference \(y_2 - y_1\) is the change in the y-values, and \(x_2 - x_1\) is the change in the x-values.

• If \(m > 0\), the line slopes upwards; a positive slope. Imagine climbing a hill.
• If \(m < 0\), the line slopes downwards; a negative slope. This is like descending a hill.
• If \(m = 0\), the line is horizontal and flat.
• If the slope is undefined, the line is vertical.

The slope is critical in determining how variables change in relation to one another. In our temperature conversion problem, a slope of \(1.8\) indicates that for every degree increase in Celsius, the Fahrenheit temperature increases by 1.8 degrees.

Coordinate Plane

A coordinate plane is a two-dimensional space where we can graph points, lines, and curves. Picture a giant sheet of graph paper. This plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Where they intersect is called the origin, marked as (0, 0).
Each point on the plane is given as an ordered pair \((x, y)\). The x-value tells us how far left or right to go, while the y-value tells us how far up or down to move.

• The x-axis divides the plane into top and bottom halves.
• The y-axis divides it into left and right halves.

It's helpful for visualizing the relationship between two variables. When we plotted the temperature points \((0, 32)\) and \((100, 212)\), we saw how the line connecting them shows the conversion from Celsius to Fahrenheit, making the abstract equation easier to understand.

Point-Slope Form

The point-slope form of a linear equation is a handy way to write an equation when you know the slope of the line and one point on the line. It looks like this:\[y - y_1 = m(x - x_1)\]Here, \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.
This form is particularly useful when you're given a problem like our temperature conversion, where the relationship between Celsius and Fahrenheit can be expressed using the slope we've calculated (\(1.8\)) and a specific point like \((0, 32)\).

• This method quickly leads to an equation by plugging in \(m\), \(x_1\), and \(y_1\).
• It shows how much \(y\) changes for a change in \(x\), starting from a known point.

Seeing the calculations and then the equation transformations helps clarify how point-slope form is directly linked to the graph and real-life scenarios.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express a line equation, easily helping to identify the slope and the y-intercept. Its format is:\[y = mx + b\]Here, \(m\) represents the slope, and \(b\) is the y-intercept, or where the line crosses the y-axis when \(x = 0\).
Converting the equation from point-slope form to slope-intercept form is straightforward:

• First, expand the equation from point-slope form \(y - 32 = 1.8(x - 0)\).
• Then, rearrange to isolate \(y\): \(y = 1.8x + 32\).

With this form, it's very clear how changes in \(x\) affect \(y\) because of \(m\), and where the line intersects the y-axis at \(b = 32\) tells the initial value when \(x = 0\). It's practical in many applications like plotting graphs, predicting outcomes, or solving real-life problems involving linear relationships.