Problem 49 - College Algebra

Question

Determinants are used to write an equation of a line passing through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$ \left|\begin{array}{ccc}x & y & 1 \\\x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1\end{array}\right|=0 $$ Use this information to work. Use the determinant to write an equation of the line passing through \((3,-5)\) and \((-2,6) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.

Step-by-Step Solution

Answer

Using the formula of determinants, the equation of the line passing through the points (3,-5) and (-2,6) is \(y = -x - 10\) in slope-intercept form.

Step 1: Substitute the Given Coordinates into the Determinant

Substitute \((x_1, y_1) = (3, -5)\) and \((x_2, y_2) = (-2, 6)\) into the determinant. This gives:\[\begin{{vmatrix}}x & y & 1 \3 & -5 & 1 \-2 & 6 & 1\end{{vmatrix}}= 0\]

Step 2: Expand the Determinant to a Polynomial

Next, use the Rule of Sarrus to find the determinant of a \(3 \times 3\) matrix. The Rule of Sarrus is as follows: 1) Multiply the elements in the main diagonal and the diagonals that arise by wrapping around the matrix. Add these products.2) Multiply the elements in the opposite diagonals (right to left diagonals, wrapping around as before). Subtract these products from the sum before.Apply the Rule of Sarrus to our matrix:\[x(-5)*1 + y(1) * 1 + 1(3) * 6 - 1(-5)*(-2) - 1*1*(-2) - x(1)*6 = 0\]

Step 3: Simplify the Expression

Simplify the above expression to get the equation of the line in general form:\[-5x + y + 18 -10 +2 + 6x = 0\]Combine like terms to rewrite the equation as:\[x + y + 10 = 0\]

Step 4: Convert to Slope-Intercept Form

Solve the equation for \(y\) to transform it to slope intercept form (\(y = mx + c\)). It becomes:\[y = -x - 10\]

Key Concepts

Linear Equation

A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. It has the general form of \(Ax + By = C\), where \(x\) and \(y\) are variables and \(A\), \(B\), and \(C\) are constants. Linear equations are fundamental in mathematics as they describe relationships between two variables directly proportional to each other. They can be used to model real-world scenarios where a constant rate of change exists.
To understand linear equations more deeply, consider the concept of slope, which indicates the steepness of the line. The slope is the ratio of the vertical change to the horizontal change between two points on the line. This characteristic helps in understanding how one variable changes in relation to another.

Slope-Intercept Form

In mathematics, the slope-intercept form is a way of writing the equation of a line so it clearly reveals the slope and the y-intercept. This form is expressed as \(y = mx + c\), where:

• \(m\) represents the slope of the line
• \(c\) is the y-intercept, the point where the line crosses the y-axis

This form is particularly useful because it allows you to quickly see the slope and y-intercept, making graphing easier. To find the slope-intercept form from a given line's equation, you can solve for \(y\) and rearrange the equation accordingly.
Transforming an equation into slope-intercept form is incredibly helpful in comparing different lines and understanding their relations. You can easily determine if two lines are parallel (if their slopes are equal) or identify potential intersections.

Matrix Expansion

Matrix expansion is the process of simplifying a matrix determinant into a polynomial form, which can then be used to solve for unknown variables. A matrix is a rectangular arrangement of numbers or expressions in rows and columns, such as the \(3 \times 3\) matrix we have in the determinant problem.
To perform matrix expansion, use methods like cofactor expansion or the Rule of Sarrus. These techniques break down the matrix into simpler expressions, allowing for easier calculation of its determinant. Once expanded, the matrix equation can be simplified to reveal important information about the set of solutions it represents.
In the context of our exercise, expanding the determinant allows you to derive the linear equation representing the line passing through two distinct points on the plane.

Rule of Sarrus

The Rule of Sarrus is a straightforward method for calculating the determinant of a \(3 \times 3\) matrix. It simplifies the process by providing a pattern to follow without needing more complex determinant calculations. This rule is specifically applicable to \(3 \times 3\) matrices and involves:

• Multiplying and adding the products of the diagonals from the top left to bottom right, including wrapping around the matrix
• Subtracting the products of the diagonals from the bottom left to top right (also including wrapping)

Applying the Rule of Sarrus to a matrix allows for quick calculation of its determinant, which can be especially useful in solving problems involving linear equations and their representations. Understanding and employing the Rule of Sarrus in exercises can save time and reduce errors, making this method an essential mathematical skill.