Problem 61 - Algebra 1

Question

Use a graph to determine whether the given three points seem to lie on the same line. If they do, prove algebraically that they lie on the same line and write an equation of the line. $$ (-3,-1),(0,1),(12,9) $$

Step-by-Step Solution

Answer

By visual inspection and algebraic confirmation, points (-3,-1), (0,1), and (12,9) lie on the line given by the equation \(y = \frac{5}{4}x + 1\).

Step 1: Visual Inspection

Plot the three points (-3,-1), (0,1), and (12,9) on a graph. Based on the graph, you may be able to visually confirm if the points lie on the same line.

Step 2: Calculate Slopes

Calculate the slope between the pairwise combinations of points: (-3,-1), (0,1) and (0,1), (12,9). Use the formula to calculate slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If all the slopes are equal, it suggests these points are on the same line.

Step 3: Algebraic Proof

Use the third point (12,9) to confirm if it lies on the line defined by the other two points. Substitute coordinates of the third point in the equation and confirm the equality.

Step 4: Equation of the Line

If the three points are confirmed to be on the same line, the equation of the line can now be written. For this, use the slope calculated in step 2 and coordinates of any point. The equation of the line in slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept.

Key Concepts

Slope Calculation

To determine if three points are collinear, we can start by calculating the slope between each pair of points. The slope is a measure of how steep a line is, given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (-3,-1) and (0,1), let's calculate the slope:

• Choose (-3,-1) as \((x_1, y_1)\) and (0,1) as \((x_2, y_2)\).
• Substitute into the formula: \( m = \frac{1 - (-1)}{0 - (-3)} = \frac{2}{3} \).

Now, calculate the slope between the next pair, (0,1) and (12,9):

• Choose (0,1) as \((x_1, y_1)\) and (12,9) as \((x_2, y_2)\).
• Substitute into the formula: \( m = \frac{9 - 1}{12 - 0} = \frac{8}{12} = \frac{2}{3} \).

Both slopes are equal, suggesting all three points could line up perfectly.

Equation of a Line

Once we've confirmed that the slopes are consistent, we can find the equation of the line that passes through these points. The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. With our calculated slope of \( \frac{2}{3} \), we can use any of the given points, like (0,1), to find \( b \).

• Substitute the point (0,1) into the equation: \( 1 = \frac{2}{3}(0) + b \)
• Simplifying, we find that \( b = 1 \)

Thus, the equation of the line is \( y = \frac{2}{3}x + 1 \). This equation represents the straight path passing through all three points if they are truly collinear.

Graphical Analysis

Plotting the points on a graph can give quick insights into whether they form a line. This visual method isn't mathematically rigorous but serves as an excellent first check. Place the points (-3,-1), (0,1), and (12,9) on a coordinate grid with correctly scaled axes. When observing the graph:

• If the points form a straight diagonal path without needing adjustment of the graph's scale, they likely are collinear.
• A straight edge or digital graphing tool can verify their alignment.

Graphical analysis is valuable for reinforcing algebraic findings, providing an intuitive understanding of why collinear points behave as they do.

Algebraic Proof

To prove algebraically that the points are on the same line, choose one point and check if it satisfies the line equation derived from the other two. Here, we can pick the point (12,9). Check if it satisfies \( y = \frac{2}{3}x + 1 \).

• Substitute: \( 9 = \frac{2}{3}(12) + 1 \)
• Calculate: \( 9 = 8 + 1 \)

Since this equation holds true, it confirms that (12,9) is on the line. This step solidifies our argument that all three points, (-3,-1), (0,1), and (12,9), lie on a common straight line, thereby concluding the algebraic proof of collinearity.