Question

Use a determinant to find an equation of the line passing through the points. $$ (-1,2),(5,3) $$

Short answer

The equation of the line passing through the points is \(y = \frac{1}{6}x + \frac{13}{6}\).

Step-by-step solution

Calculate the Slope

First, calculate the slope \(m\) of the line using the formula \[ m = \frac{y2 - y1}{x2 - x1} \] Substitute the given points into the formula, we get: \[ m = \frac{3 - 2}{5 - (-1)} = \frac{1}{6} \]

Determine the y-intercept

Next, substitute one of the points and the slope into the equation \(y = mx + c\) to find 'c'. Let's use the point (-1, 2) as an example, we get: \[ 2 = \frac{1}{6}*(-1) + c \] Solving this gives \(c = \frac{13}{6}\).

Formulate the Line Equation

Finally, substitute the slope and the y-intercept into the equation \(y = mx + c\) to find the equation of the line. We get: \[ y = \frac{1}{6}x + \frac{13}{6} \].

Key concepts

Slope Calculation

Determining the slope of a line is a foundational skill in algebra that facilitates understanding of linear relationships. The slope indicates how steep a line is and the direction in which it tilts. To compute the slope, or the rate of change, we use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, the variables \( (x_1, y_1) \) and \( (x_2, y_2) \) represent two distinct points on a line. In the exercise, these points are \( (-1, 2) \) and \( (5, 3) \). Upon substituting these values into the slope formula, the slope \( m \) is calculated to be \( \frac{1}{6} \). This value tells us that for every one-unit increase in 'x', the value of 'y' increases by \( \frac{1}{6} \) units. Slope calculation not only helps in defining the line equation but also assists in analyzing the line's behavior on a graph.

Determinants in Algebra

While the term 'determinant' is mostly associated with matrices in algebra, its application extends to various aspects of linear equations and higher-dimensional geometry. In this context, considering a determinant as a tool may be a metaphor for the decisive factor that helps establish certain values like slope or intercepts in line equations. It's important not to confuse the notion of 'determinants' as used informally in this exercise with the formal definition involving matrices. For students seeking a better grasp, it's crucial to understand that the determinant of a matrix is a specific scalar value that can be computed from the elements of a square matrix and has several applications, including solving systems of linear equations, finding the inverse of a matrix, and calculating areas or volumes.

Y-intercept Calculation

The y-intercept of a line is the point where the line crosses the y-axis. It is a critical component of the line equation and can be found using the formula \( y = mx + c \) where 'c' represents the y-intercept. After the slope 'm' has been calculated, you can choose any point on the line to solve for 'c'.
For instance, using the point \( (-1, 2) \) from the exercise and the calculated slope \( \frac{1}{6} \), the equation to uncover 'c' would be:
\[ 2 = \left(\frac{1}{6}\right)(-1) + c \]
This would yield \( c = \frac{13}{6} \), placing the y-intercept at \( \frac{13}{6} \) on the y-axis. Thus, when graphing a line or understanding its equation, the y-intercept offers an essential starting point and easier visualization of the linear relationship depicted by the equation.