Problem 44 - Algebra 1

Question

Write an equation of the line that passes through the points. (1,4),(3,7)

Step-by-Step Solution

Answer

The line equation is \(y = 1.5x + 2.5\)

Step 1: Calculate the Slope

The slope of the line that passes through the points (1,4) and (3,7) is calculated using the formula \((y_2 - y_1) / (x_2 - x_1)\). Therefore, the slope \(m = (7 - 4) / (3 - 1) = 1.5\)

Step 2: Use the slope-intercept form to find the equation of the line

We know the slope-intercept form of a line is \(y = mx + c\). We substitute the calculated slope ('m') into the equation. Now we have \(y = 1.5x + c\). We can use either of the points given to find 'c'. If we take the point (1,4), by substituting these values into the equation, we will have \(4 = 1.5*1 + c\), and from this, we can solve for c to get \(c = 4 - 1.5 = 2.5\)

Step 3: Write the equation of the line

Substituting the calculated slope and y-intercept into the slope-intercept form equation: The equation of the line that passes through the points (1,4) and (3,7) is \(y = 1.5x + 2.5\)

Key Concepts

Slope Formula

The slope formula is a fundamental aspect of coordinate geometry that defines the steepness of a line. It is the ratio of the vertical change (\textdelta y) to the horizontal change (\textdelta x) between two distinct points on a line. Formally, if you have two points \text{(x_1, y_1)} and \text{(x_2, y_2)}, the slope m is calculated as follows:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]
Consider the points in our exercise, \text{(1,4)} and \text{(3,7)}. By applying the slope formula, we have:
\[ m = \frac{(7 - 4)}{(3 - 1)} = \frac{3}{2} \]
The slope of 1.5 reflects how much the line rises vertically for each unit of horizontal movement. Understanding how to calculate slope is critical for graphing linear equations and analyzing the relationship between variables in various fields such as physics, economics, and social sciences.

Slope-Intercept Form

The slope-intercept form of a line is one of the most commonly used representations of a linear equation, given by the formula \( y = mx + c \). In this expression, \text{'m'} denotes the slope of the line, and \text{'c'} represents the y-intercept, the point where the line crosses the y-axis.

Knowing the slope and any single point on the line allows us to find the y-intercept. For example, using our earlier calculated slope of 1.5 and the point \text{(1,4)}, we can find the value of \text{'c'}:
\[ 4 = 1.5(1) + c \]
Hence, \text{'c'} equals 2.5, demonstrating that the intercept of our line on the y-axis is at the point \text{(0,2.5)}. This ability to compose the equation of a line quickly is incredibly advantageous, not just in mathematics, but also in various practical applications like economics to predict costs or in physics to understand velocity.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points, lines, and figures in a two-dimensional plane using coordinates. It is pivotal in our exercise, as it helped us leverage the coordinates of two given points to obtain the equation of a line.

Through this system, we can perform numerous calculations, such as finding distances, midpoints, slopes, and equations of lines—all essential tools in solving more complex geometric problems. It's not just limited to the classroom but is also extensively applied in fields such as navigation, computer graphics, and robotics, where spatial relationships are crucial.

Linear Equations

Linear equations form the foundation for understanding relationships between two variables in algebra. They are so named because their graphs are straight lines. A linear equation in two variables, \text{x} and \text{y}, takes the general form \( y = mx + c \), where again, \text{'m'} is the slope and \text{'c'} is the y-intercept.

These equations are not just academic exercises; they model a multitude of real-world phenomena, from calculating simple interest to predicting population growth. By mastering linear equations, students are better equipped to understand and interact with the quantitative world around them, making this mathematical concept one of the most valuable across various scientific and practical disciplines.