Problem 16 - Algebra 1: Concepts and Skills

Question

Plot the points and draw a line that passes through them. Use the rise and run to find the slope. \((2,2)\) and \((6,-1)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((2,2)\) and \((6,-1)\) is \(-0.75\), which means the line will go down by 3 units for a 4 units run to the right.

Step 1: Find the rise and run

The rise is the difference in the y-coordinates of the two points \((y2 - y1)\), and the run is the difference in the x-coordinates of the two points \((x2 - x1)\). Here, the coordinates of the two points are \((2,2)\) and \((6,-1)\). Therefore, the rise is \((-1 - 2) = -3\) and the run is \((6 - 2) = 4\).

Step 2: Calculate the slope

The slope of a line given two points is calculated as the ratio of rise to run, or \(-3 / 4 = -0.75\). So, the slope of the line is \(-0.75\).

Step 3: Plot the points and draw the line

The given points \((2,2)\) and \((6,-1)\) should be plotted on a graph. A straight line should then be drawn through these points. The slope of the line indicates that for every 1 unit increase in x, there will be a 0.75 unit decrease in the y direction, or for every 4 units increase in x, there will be 3 units decrease in y. This line will pass through both points and have the calculated slope.

Key Concepts

Plotting Points on a Graph

Understanding how to plot points on a graph is foundational for visualizing mathematical concepts, particularly in algebra. To begin, imagine a graph as a map. The streets running left and right represent the x-axis, and those running up and down represent the y-axis. Each point is an address, telling us exactly where to find a location on our map. For the points ewline

Slope Formula

The slope formula is critical in understanding the steepness and direction of a line on a graph. In mathematics, we define the slope ewline

Rise Over Run

The concept of 'rise over run' is the heartbeat of the slope formula, symbolizing the change in vertical position (rise) compared to the horizontal position (run). It's like the stairs in your home; with each step, you rise up while also moving forward. To find the slope, we first identify the rise by subtracting the y values of our two points. Given the points ewline

Graphing Linear Equations

Graphing linear equations is the process of drawing the representation of an equation as a straight line on a coordinate plane. This visual representation helps in understanding how two variables relate to each other. To graph a line, we typically need either an equation or two points. In our exercise, we started with two points: ewline