Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the general form of the equation of the line that passes through the points. $$(2,7),(3,10)$$

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x + 1 \)

Step 1: Calculate the slope

Firstly, let's calculate the slope (m) of the line, which is the change in 'y' divided by the change in 'x'. Given two points (x1, y1) and (x2, y2), the slope 'm' is calculated as:\[ m = \frac{y2 - y1}{x2 - x1} \]Substitute the given points into the formula, we get:\[ m = \frac{10 - 7}{3 - 2} = \frac{3}{1} = 3 \]

Step 2: Determine the y-intercept

Now, let's find the y-intercept (c).We know the equation of a line is \[ y = mx + c \]which is y = slope * x + y-intercept. We can rearrange the equation to get \[ c = y - mx \]Substitute x=2, y=7 and m=3 from the given point or the slope into the formula, we have:\[ c = 7 - 3 * 2 = 7 - 6 = 1 \]

Step 3: Write the equation of the line

Now that we have the slope and the y-intercept, we can substitute those values into the equation of a line to get the required line equation:\[ y = 3x + 1\]

Key Concepts

Slope Calculation

Understanding the slope of a line is essential when dealing with linear equations. Slope represents the rate at which the line rises or falls as you move from left to right across the graph. To calculate the slope, often denoted as 'm', you need two points on the line, let's say point A \( (x_1, y_1) \) and point B \( (x_2, y_2) \). The formula for the slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, if you're given the coordinates \( (2, 7) \) and \( (3, 10) \), the slope calculation would look like this:\[ m = \frac{10 - 7}{3 - 2} \]
Which simplifies to \(m = 3\). This means that for every one unit you move to the right on the graph, you move up three units. It's crucial to get this step correct, as the slope is a key component in the linear equation of a line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. It is typically represented by the letter 'c'. The y-intercept is an integral part of the line's equation, as it helps to locate the entire line on the graph. Once you have the slope from the slope calculation, use the formula \( c = y - mx \) to find the y-intercept.
Let's say you have a slope, 'm', of 3, and you are using one of the points \( (2, 7) \) from the line. By inserting these into the formula, you get:\[ c = 7 - (3 \times 2) \]
This simplifies to \(c = 1\). Therefore, the y-intercept (the point where the line crosses the y-axis) is at \( (0, 1) \). Identifying the y-intercept is a step that shouldn't be overlooked as it describes the starting point of your line when graphing.

Linear Equations

Linear equations are the mathematical representations of straight lines. The most common form of a linear equation is the slope-intercept form, given as \( y = mx + c \). Here, 'm' stands for the slope and 'c' represents the y-intercept. This form provides a straightforward method for graphing the line as you can see the slope and y-intercept at a glance.
With the slope \(m = 3\) and y-intercept \(c = 1\), as previously calculated, you would write the linear equation of the line that passes through the points \( (2, 7) \) and \( (3, 10) \) as:\[ y = 3x + 1 \]
This formula allows you to compute the y-value for any given x-value on this particular line, showing the linear relationship between the x and y variables. The process of forming the equation with the correct slope and y-intercept is crucial, so take your time to understand each part well.