Problem 64 - 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. $$ (3,-3),(-1,13),(1,5) $$

Step-by-Step Solution

Answer

By plotting the points and calculating the slopes, it can be algebraically confirmed that the points (3, -3), (-1, 13) and (1, 5) do lie on the same line. The equation of the line that includes these three points can be determined using the point-slope formula.

Step 1: Plotting the Points

Plot the points (3, -3), (-1, 13) and (1, 5) on a graph. This will help to visually determine if the points could lie on the same line.

Step 2: Visual Inspection

Visually inspect the graph. If these points seem to fall on the same line, you can proceed to prove it algebraically. If they don't seem to lie on the same line, they probably don't, but that too would need to be verified algebraically. In this case, they do seem to lie on the same line based on visual inspection.

Step 3: Algebraic Verification

Using the formula for the slope of a line \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) applied to pairs of points: Calculate the slope between point 1 and 2, and between point 2 and 3. The points will lie on the same line if and only if these slopes are equal. If \(m_{12} = m_{23}\), then the points lie on the same line.

Step 4: Finding Equation of the Line

If it is confirmed that the points lie on the same line, use the point-slope formula to find the equation of this line. The point-slope formula is \(y-y_{1}=m(x-x_{1})\). Any of the given points can be used in place of (x_1, y_1). Use the confirmed slope for m, in the formula to get the equation of the line.

Key Concepts

Plotting Points

Understanding how to plot points on a coordinate plane is fundamental in algebra. Every point on a plane is defined by an ordered pair of numbers, known as coordinates. The first number, or the x-coordinate, specifies the horizontal position, while the second number, the y-coordinate, specifies the vertical position.

To plot the points (3, -3), (-1, 13), and (1, 5), you start by locating the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis. For example, for the point (3, -3), you will move three units to the right on the x-axis, because the x-coordinate is positive, and then three units down, because the y-coordinate is negative. After plotting all points, if they appear to form a straight line when connected, they may indeed lie on the same line.

Slope of a Line

The slope of a line is a measure of its steepness and direction and is denoted by the letter 'm'. Mathematically, it is expressed as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. To calculate the slope (mtype>, using the formula y_{2 }- y_{1 }} {x_{2} - x_{1}}), you subtract the y-coordinate of the first point from the y-coordinate of the second point and then divide this by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point.

For instance, using the given points (3, -3) and (-1, 13), the slope is calculated as: 13 - (-3 decision}} perhaps {-1} - 3)), which simplifies to 4}. If the slope between each pair of points is identical, it suggests that the points lie on the same line.

Point-Slope Formula

After determining that three points lie on the same line, the next step involves finding the line's equation. This can be achieved by using the point-slope formula, an essential concept in coordinate geometry. It is written as y-y_{1} = m (x-x_{1})m=' and (x_{1}, y_{1})appears to be a specific point on the line. The beauty of this formula is its versatility—you can use any point on the line to establish the line's equation.

To apply the point-slope formula, plug in the slope you've calculated and any of the points' coordinates as (x_{1}, y_{1})). By doing so, you'll generate the particular equation of the line that passes through the points.

Algebraic Verification

Even though plotting points and visually inspecting them provides a good initial indication that they fall on the same line, algebraic verification is what ultimately confirms this. It entails calculating the slope from one point to another and comparing. Only if the slopes are identical is it algebraically proven that the points are collinear.

In our context, we computed the slopes between (3, -3)) and (-1, 13), and between (-1, 13)) and (1, 5)) and found them to be the same, confirming that our points indeed lie on the same straight line. This algebraic validation gives the foundational proof necessary to conclude the points' alignment without merely relying on a graphical representation.