Problem 21 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (-1,-2),(2,6) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-1,-2) and (2,6) is \(y = (8/3)x + 2/3\).

Step 1: Plot the Points

Begin by plotting the two given points on a graph. Typically, the Cartesian Coordinate system is used, with x-axis and y-axis. Each point has two coordinates, with (-1,-2) denoting a point 1 unit to the left of the origin and 2 unit below, and the point (2,6) 2 units to the right and 6 units up.

Step 2: Find the Slope

The slope between two points (x1, y1) and (x2, y2) is given by the formula \((y2 - y1) / (x2 - x1)\). Substituting the given points: (2,6) and (-1,-2) into this equation results in \((6 - (-2)) / (2 - (-1))\) which simplifies to \(8 / 3\). So the slope \(m\) is \(8/3\).

Step 3: Find the y-intercept

The slope intercept form of a line is \(y = mx + b\), where \(b\) is the y-intercept. With already known slope \(m = 8/3\) and a known point on the line (2,6), the y-intercept can be found by rearranging this formula to \(b = y - mx\) and substiting the current values. This results in \(b = 6 - (8/3)*2 = 6 - 16/3 = 18/3 - 16/3 = 2/3\). Therefore, the y-intercept \(b\) is \(2/3\).

Step 4: Write the equation of the line

Having calculated the slope \(m = 8/3\) and the y-intercept \(b = 2/3\) you can now write down the equation of the line in slope intercept form \(y = mx + b\). The final equation for the line therefore is \(y = (8/3)x + 2/3\).

Key Concepts

Graphing Points

When you're starting with graphing points, it's all about understanding the coordinates. Each point on a graph is like a set of directions. The first number shows how far to move along the x-axis (horizontal), and the second number tells you how far to move along the y-axis (vertical). Let's take the points

• (-1, -2): Begin at the origin (0,0). Move 1 unit to the left, then 2 units down.
• (2, 6): Start at the origin, move 2 units to the right, then 6 units up.

Once both points are plotted, draw a straight line through them. This line represents all the solutions to the equation we'll write soon.
Graphing ensures you have a visual reference, making it easier to understand how the equation connects these points.

Slope Calculation

The slope of a line is a numerical measure that describes its steepness and direction. It's calculated as the 'rise' (change in y) over the 'run' (change in x). For the points (-1,-2) and (2,6):

• Find the change in y:
  \(y_2 - y_1 = 6 - (-2) = 8\)
• Find the change in x:
  \(x_2 - x_1 = 2 - (-1) = 3\)

This results in a slope of \(\frac{8}{3}\).
Slope determines how fast y values increase as x increases. A positive slope means an uphill line (bottom-left to top-right), ensuring that as x increases, so does y.
Always use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for consistent results.

Y-Intercept

The y-intercept is where the line crosses the y-axis and occurs when x is 0. It's represented by the 'b' in the equation \(y = mx + b\). To find it:

• Use one of the points, e.g., (2,6), and the slope which you've calculated as \(\frac{8}{3}\).
• Rearrange the formula to solve for b:
  \( b = y - mx = 6 - \left(\frac{8}{3}\right) \times 2 = 6 - \frac{16}{3} = \frac{18}{3} - \frac{16}{3} = \frac{2}{3} \)

Now, the y-intercept is \(\frac{2}{3}\).
With the slope and y-intercept identified, you can write the line's equation in slope-intercept form as \(y = \frac{8}{3}x + \frac{2}{3}\).
This form makes it easy to immediately see the slope and where the line crosses the y-axis.