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Chapter 8: Problem 43

Use a determinant to find an equation of the line passing through the points. $$(0,0),(5,3)$$

Short Answer

Expert verified
The equation of the line passing through the points (0,0) and (5,3) is \(y = \frac{3}{5}x\)

Step by step solution

01

Calculate the Slope

First, use the slope formula to find the slope \(m\) of the line. The formula is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). With given points (0,0) and (5,3), this becomes \(m = \frac{3 - 0}{5 - 0} = \frac{3}{5}\). So, the slope \(m\) of the line is \(\frac{3}{5}\).
02

Calculate the Y-Intercept

Since the line passes through the origin (0,0), the y-intercept \(c\) is 0.
03

Formulate Equation of the line

The general equation of a line is in the form \(y = mx + c\). Substituting the values of slope \(m = \frac{3}{5}\) and y-intercept \(c = 0\), we get the equation of the line as \(y = \frac{3}{5}x + 0\). This simplifies to \(y = \frac{3}{5}x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant
In mathematics, the determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix and can be used in various calculations.
  • A determinant helps in revealing if a system of equations has a unique solution (non-zero determinant) or not (zero determinant).
  • In the context of finding the equation of a line through two points, the determinant is often leveraged to check linear dependence.
In coordinate geometry, when determining the equation of a line through specific points, determinants can theoretically confirm if three points are collinear. While not used directly to find the slope or y-intercept, determinants offer a deeper understanding of the relationships between points on a plane.
Equation of a Line
The equation of a line represents a straight line on a graph. The most common form is the slope-intercept form: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept. This formula is essential for graphing and understanding linear functions.

The equation consists of:
  • **Slope \( m \):** Indicates the steepness and direction of the line.
  • **Y-intercept \( c \):** The point where the line crosses the y-axis.
For the points given in the original exercise, \((0,0)\) and \((5,3)\), these components enable us to formulate the specific equation of the line as \( y = \frac{3}{5}x \). No additional constant is needed as the line passes through the origin.
Slope
The slope of a line is a measure of its steepness or inclination. Mathematically, the slope \( m \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the change in the y-coordinates divided by the change in the x-coordinates between two points.

In this context:
  • For points \((0,0)\) and \((5,3)\), the slope calculates as \( \frac{3}{5} \), indicating that for every five units the line moves horizontally, it moves three units vertically.
  • Positive slope means the line is inclined upwards, while a negative slope indicates a downward inclination.
This concept is crucial for understanding how to graph lines and interpret linear equations.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. The y-intercept is denoted by \( c \) in the line equation \( y = mx + c \).
  • In our exercise, the y-intercept is found by substituting \( x = 0 \) into the line's equation.
  • For the given points \((0,0)\) and \((5,3)\), since the line passes through the origin, the y-intercept \( c \) is 0.
The y-intercept is important because it gives us a starting point for drawing our line when graphing on a coordinate plane. Understanding this concept helps in quickly identifying the line's position regarding the y-axis when interpreting graphs.

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Most popular questions from this chapter

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrrr}1 & -2 & -1 & -2 \\\3 & -5 & -2 & -3 \\\2 & -5 & -2 & -5 \\\\-1 & 4 & 4 & 11\end{array}\right]$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l}4 x-2 y+3 z=-2 \\\2 x+2 y+5 z=16 \\\8 x-5 y-2 z=4\end{array}\right.$$

At this point in the text, you have learned several methods for finding an equation of a line that passes through two given points. Briefly describe the methods that can be used to find the equation of the line that passes through the two points shown. Discuss the advantages and disadvantages of each method.

Solve for \(x\). $$\left|\begin{array}{rr} x+3 & 2 \\ 1 & x+2 \end{array}\right|=0$$

Use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. $$\left\\{\begin{aligned} 2 x+y-z+2 w &=-6 \\ 3 x+4 y &+w=1 \\ x+5 y+2 z+6 w &=-3 \\ 5 x+2 y-z-w &=3 \end{aligned}\right.$$
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