Show me what y=5x+6 looks like on a graph and how to find rise over run
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Let's explore the linear equation y equals 5x plus 6. This is written in slope-intercept form, where y equals mx plus b. In our equation, the slope m equals 5, which tells us how steep the line is. The y-intercept b equals 6, which is where the line crosses the y-axis. Understanding these components will help us graph the line and calculate rise over run.
Now let's plot the graph of y equals 5x plus 6. First, we plot the y-intercept at point (0, 6). Next, we use the slope of 5, which means for every 1 unit we move right, we move up 5 units. Starting from (0, 6), moving right 1 and up 5 gives us (1, 11). Continuing this pattern gives us (2, 16). Going in the opposite direction from the y-intercept, we get (-1, 1) and (-2, -4). Finally, we connect all these points to form a straight line.
Now let's understand rise over run. Rise over run is a way to calculate the slope of a line. Rise refers to the vertical change, or how much the y-value changes. Run refers to the horizontal change, or how much the x-value changes. Looking at our line, let's use points (0, 6) and (1, 11). The run is the horizontal distance from 0 to 1, which equals 1. The rise is the vertical distance from 6 to 11, which equals 5. Therefore, the slope equals rise divided by run, which is 5 divided by 1, equals 5.
Let's calculate rise over run step by step using points (0, 6) and (2, 16). First, we find the rise by subtracting the y-coordinates: 16 minus 6 equals 10. Next, we find the run by subtracting the x-coordinates: 2 minus 0 equals 2. Finally, we calculate the slope by dividing rise by run: 10 divided by 2 equals 5. Notice that this matches our original slope value of 5 from the equation y equals 5x plus 6.
Let's verify that rise over run gives the same slope regardless of which points we choose. First, using points (-1, 1) and (1, 11): the rise is 11 minus 1, which equals 10, and the run is 1 minus negative 1, which equals 2. So the slope is 10 divided by 2, equals 5. Now using points (-2, -4) and (0, 6): the rise is 6 minus negative 4, which equals 10, and the run is 0 minus negative 2, which equals 2. Again, the slope is 10 divided by 2, equals 5. This demonstrates that the slope is constant for any two points on a straight line, confirming our equation y equals 5x plus 6.