In the slope-intercept equation y = mx + b), the slope m is multiplied on the x value and b is the y-intercept, where the line crosses the y-axis. Graphing from this format can be rather easy, particularly if m and b are relatively simple numbers rather than fractions or complicated decimals. But, let's a look at the meanings that slope and y-intercept can have in solving word problems.
You've learned that slope defines the ratio of the “rise” (in y) divided by the “run” (in x) between two points on a line, in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. For instance, in y = ( 2x/3 ) + 2, the slope is m = ⅔. This means that, from any point on this line, you can get to another point on the line by going up 2 units then going to the right 3 units.
But we could also view this slope as a fraction over 1:
The ratio remains the same but, for every one unit that x moves to the right, y goes up by two-thirds of a unit. While this doesn't graph as easily as m = two-thirds, it can be a more useful way of viewing things when doing word problems.
Often, word problems deal with changes over time. For instance, a problem might deal with how tall a plant grows if it increases by a fixed amount every three months.
When x = 0, the corresponding y-value is the y-intercept. However, in the context of word problems, the y-intercept also refers to the starting value. In the equation y = ( 2x/3 ) + 2, the y-intercept, 2, would be the size of the plant when we started keeping track of the data and the slope would indicate its rate of growth. Let's do a few examples, to see how this works:
The average lifespan of American women has been tracked, and the model for the data is y = 0.2t + 73, where t = 0 corresponds to 1960. Explain the meaning of the slope and y-intercept.
The slope is m = 0.2 and means that every year, the average lifespan of American women increased by 0.2 years.
The y-intercept = 73 and means that, when they started keeping track in 1960, the average lifespan was 73 years.
The equation for the speed of a ball that is thrown straight up in the air is given by v = 128 − 32t, where v is the velocity in feet per second and t is the number of seconds after the ball is thrown. What is the meaning of the slope and with what initial velocity was the ball thrown?
The slope is m = −32 and means that every second the speed decreases by 32 feet per second.
The y-intercept = 128 and means that the ball was launched at 128 feet per second.
Fishermen in the Great Lakes have been recording the dead fish they encounter while fishing in the region. The Department of Environmental Services monitors the pollution index for the Great Lakes and their model for the number of fish deaths y for a given pollution index x is y = 9.607x + 111.958. What are the meanings of the slope and y-intercept?
The slope of m = 9.607 and means that, for every increase of one unit in the pollution index there are approximately nine or ten more fish deaths during the year.
The y-intercept of 111.958 means that, even if the pollution index is zero there would still be about 112 fish deaths a year.
Word problems with linear equations almost always work this way: the slope is the rate of change and the y-intercept is the starting value.
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