Problem 80 - Algebra 2

Question

Points that are on the same line are collinear. Use the definition of slope to determine whether the given points are collinear. $$ (3,-5),(-3,3),(0,2) $$

Step-by-Step Solution

Answer

The slopes between the pairs of points are different (-4/3 and -1/3), therefore the points (3,-5), (-3,3), and (0,2) are not collinear.

Step 1: Calculate the slope between the first two points

To find the slope between the first two points (3,-5) and (-3,3), use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The slope calculation will be \( m_1 = \frac{3 - (-5)}{-3 - 3} \).

Step 2: Calculate the first slope

Performing the subtraction in the numerator and denominator gives us \( m_1 = \frac{3 + 5}{-3 - 3} = \frac{8}{-6} = -\frac{4}{3} \).

Step 3: Calculate the slope between the second and third points

Now, calculate the slope using the second and third points, (-3,3) and (0,2), with the same slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The calculation is \( m_2 = \frac{2 - 3}{0 - (-3)} \).

Step 4: Calculate the second slope

Perform the subtraction to get \( m_2 = \frac{2 - 3}{0 + 3} = \frac{-1}{3} \).

Step 5: Compare the two slopes

Compare the slopes \( m_1 \) and \( m_2 \). If the slopes are equal, then the points are collinear. In this case, \( m_1 = -\frac{4}{3} \) and \( m_2 = -\frac{1}{3} \). Since they are not equal, the points are not collinear.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in geometry and algebra, as it measures the steepness and direction of the line. Slope is represented by the letter 'm' and can be thought of as the 'rise over run' between any two points on a line.

When visualizing the slope, imagine you're walking on a path. The 'rise' is the vertical change from one point to another, and the 'run' is the horizontal change. If you move up on the y-axis, the rise is positive, and if you move down, it's negative. Similarly, moving right on the x-axis gives a positive run, while moving left gives a negative one.

The magnitude of the slope indicates the steepness: higher absolute values mean steeper slopes. The sign indicates the direction: positive slopes go upwards from left to right, and negative slopes go downwards. A zero slope means the line is horizontal, and an undefined slope (division by zero in the slope formula) indicates a vertical line.

Determining Collinearity

Collinearity is the property of points lying on the same straight line. Determining whether points are collinear involves checking if the slope between any two pairs of points is the same. If all pairs of points have the same slope, they are said to be collinear, as they lie on one straight line.

To visualize this, imagine connecting two pairs of points with a piece of string. If the strings overlap completely, the points are collinear. If they don't, they lie on different lines. This method allows for a graphical check of collinearity.

Collinearity can also be determined algebraically using the slope formula. By computing the slope using pairs of points and comparing these values, you can determine whether the points share the same line. If the slopes are equal, the points are collinear; if not, they reside on different lines.

Slope Formula

The slope formula is a mathematical tool used to calculate the slope of a line when given two points. It is expressed as \[\begin{equation}m = \frac{y_2 - y_1}{x_2 - x_1}\end{equation}\], where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. This formula can be applied to any two points on a line to find the slope of that line.

When using this formula, it's essential to be consistent with the order of the coordinates to get the correct slope value. Mixing up the order can result in the wrong slope and, consequently, inaccurate conclusions about the relationship between points. Remember, the slope is a ratio; thus, it simplifies just as any other fraction would. For example, a slope of \(\frac{2}{4}\) is the same as a slope of \(\frac{1}{2}\), indicating the same line, just represented in a different form.