Learn Before
Finding the Slope from a Graph
A line is graphed on the -coordinate plane passing through the points and . To find its slope, apply the four-step rise-over-run procedure.
Step 1 — Locate two points with integer coordinates. The points and both have integer coordinates.
Step 2 — Identify the leftmost point. The point is farther to the left, so begin there.
Step 3 — Sketch a right triangle and count the rise and run. Starting at , move up to (a rise of ) and then across to (a run of ).
Step 4 — Take the ratio of rise to run:
The slope of the line is .
Reversing direction: Starting instead at and moving to , the rise is (downward) and the run is (leftward). The slope formula gives . Because both the rise and the run change sign when the starting point is reversed, the negatives cancel and the slope remains the same. This confirms that it does not matter which point is used as the starting point — the slope of the line is always identical.
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Elementary Algebra @ OpenStax
Ch.4 Graphs - Elementary Algebra @ OpenStax
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An urban planner is examining a graph of a road elevation change. To find the slope of the road from the graph, place the following steps in the correct order according to the standard procedure for calculating slope from a visual representation.
A quality control inspector is analyzing a graph of machine output over time. When using the 'rise over run' method to determine the slope of the production line, which of the following correctly describes the 'run'?
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An inventory analyst is reviewing a graph that tracks the depreciation of office equipment over time. To ensure an accurate calculation of the slope from the visual grid, the analyst follows a standard procedure where the first step is to locate two points on the line whose coordinates are ____.
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A maintenance supervisor is analyzing a graph of a building's energy consumption, which shows a steady decrease over the last week. When following the standard procedure to calculate the slope of this line, how should the supervisor record the 'rise' if the vertical distance between the two selected points is downward?
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Learn After
A construction foreman is verifying the slope of a ramp on a site plan where the line passes through the points (2, 3) and (7, 6). Arrange the following steps in the correct order to determine the slope of this line using the rise-over-run method.
A site supervisor is reviewing a blueprint for a new access ramp. The ramp's incline is plotted on a coordinate grid and passes through the points (2, 3) and (7, 6). Match each slope-related term with its corresponding value or coordinate pair based on the rise-over-run method.
A logistics coordinator is mapping a delivery route on a coordinate grid. The route follows a straight line passing through the points (2, 3) and (7, 6). According to the rise-over-run procedure, what is the slope of this route?
An HVAC technician is monitoring the pressure in a ventilation system. The pressure readings, when plotted on a coordinate grid, form a straight line that passes through the points (2, 3) and (7, 6). True or False: When using the rise-over-run method, the technician will obtain a different slope value if they begin the calculation at the point (7, 6) and move to (2, 3) instead of starting at (2, 3) and moving to (7, 6).
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A logistics coordinator is mapping a delivery route on a coordinate grid. The route follows a straight line passing through the points (2, 3) and (7, 6). When using the rise-over-run method starting at the point (2, 3), the horizontal change, or 'run,' required to reach the second point is ____ units.
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A facilities technician is plotting the incline of a drainage pipe on a coordinate grid. The line representing the pipe passes through the points (2, 3) and (7, 6). According to the rise-over-run method, what is the numerical value of the 'rise' (vertical change) between these two points?
A quality control inspector is verifying the incline of a structural component on a blueprint that passes through the points (2, 3) and (7, 6). To check for mathematical consistency using the rise-over-run method, the inspector decides to calculate the slope by starting at the point (7, 6) and moving toward the point (2, 3). According to the 'reversing direction' principle, what specific values for the 'rise' and 'run' are recorded in this instance?