Question

Determine whether the given points lie on a straight line. $$ A(-2,1), B(1,7), \text { and } C(4,13) $$

Short answer

The slopes between points A(-2,1) and B(1,7) and between points B(1,7) and C(4,13) are both equal to 2. Therefore, the points A, B, and C lie on a straight line.

Step-by-step solution

Calculate the slope between points A and B

To find the slope between points A(-2,1) and B(1,7), we use the formula: Slope (m) = \(\frac{y_2 -y_1}{x_2 - x_1}\) Using the coordinates of points A(-2,1) and B(1,7), we have: m_AB = \(\frac{7 - 1}{1 -(-2)}\)

Simplify the slope between points A and B

Now, we will simplify the slope calculated in Step 1: m_AB = \(\frac{6}{3}\) m_AB = 2

Calculate the slope between points B and C

Next, we will find the slope between points B(1,7) and C(4,13) using the same slope formula as in Step 1: m_BC = \(\frac{13 - 7}{4 - 1}\)

Simplify the slope between points B and C

Now, we will simplify the calculated slope in Step 3: m_BC = \(\frac{6}{3}\) m_BC = 2

Compare the slopes and draw a conclusion

Comparing the slopes calculated in Steps 2 and 4, we have: m_AB = m_BC = 2 Since the slopes between points A and B and between points B and C are equal, we can conclude that the given points lie on a straight line.

Key concepts

Slope Formula

Understanding the slope formula is crucial for determining the linear relationship between points on a coordinate plane. The slope indicates how steep a line is and the direction it's heading, whether upwards or downwards in relation to the horizontal axis. The formula to calculate the slope between two points, say point 1 with coordinates \( (x_1, y_1) \) and point 2 with coordinates \( (x_2, y_2) \) is given by the expression \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

This formula essentially shows how much the 'y' value (vertical change) increases or decreases with each unit of increase in the 'x' value (horizontal change). In the example given, the slope \( m_{AB} \) between points A and B, and the slope \( m_{BC} \) between points B and C is calculated using this formula. Both slopes were found to be 2, signifying that for every unit you move to the right along the x-axis, the y value increases by 2 units, indicating a consistent rise and thus confirming collinearity.

Coordinate Geometry

Coordinate geometry is an area of mathematics that represents geometric figures within a coordinate system. By employing algebraic equations, we can analyze points, lines, and shapes on the plane. Particularly for points, when determining if they lie on the same straight line, coordinate geometry simplifies the task to calculating and comparing slopes.

When we speak of collinearity in coordinate geometry, we are determining if a single straight line can pass through all given points. As we use the slope formula for this purpose, consistent slope values, as shown in the exercise by the points A, B, and C, confirm that these points are indeed collinear. If the slopes had varied, this would suggest that the points do not line up on a single straight line.

Linear Equations

Linear equations play a pivotal role in understanding the relationships between variables in coordinate geometry. They are algebraic statements that reflect a straight line when graphed on a coordinate plane and are generally presented in the form \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) the y-intercept. The slope, as previously mentioned, tells us about the angle of the line and its direction. If two linear equations have the same slope, they are either parallel or the same line.

In the context of the given exercise, had we written linear equation forms for each segment between the points A to B and B to C, we would have identified the consistent slope of 2 as the 'm' value in each equation. This reinforces the concept of collinearity, as matching slopes suggest that the line segments are extensions of one another, and together, they represent a single straight line passing through all three points.