We have seen that the slope of a line measures how much the value of y changes for every so much that the value of x changes. For instance, in the line y = ( 3/5 )x – 2, the slope is m = 3/5. This means that, starting at any point on this line, you can get to another point on the line by going up 3 units and then going to the right 5 units. But we could also view this slope as a fraction over 1:
slope is 3/5 over 1
In other words, for every one unit that x moves over to the right, y goes up by three-fifths of a unit. While this doesn't necessarily graph as easily as "three up and five over", it can be a more useful way of viewing things when you're doing word problems.
Often, linear-equation word problems deal with changes over the course of time; the equations will deal with how much something changes as time passes. For instance, an exercise might deal with how the population grows in a certain city, with the population increasing by a certain fixed amount every year.
When x = 0, the corresponding y-value is the y-intercept. In the particular context of word problems, the y-intercept (that is, the point when x = 0) also refers to the starting value; that is, the value when you started taking your reading or started "counting" the time and its changes. In the example from above, the y-intercept would be the population when the sociologists started keeping track of the population. If they started taking their measurements or doing their calculations from a "base" year of 1980, then "x = 0" would correspond to "the year 1980", and the y-intercept would correspond to "the population in 1980".
The following are a few examples that further demonstrate 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.
What is the slope? It is m = 0.2. This values tells me that, for every increase of 1 in my input variable t (that is, for every increase of one year), the value of my output variable y will increase by 0.2. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
What is the meaning of the slope? It means that, every year, the average lifespan of American women increased by 0.2 years, or about 2.4 months.
When t = 0, what is the value of y? Looking at the equation, I see that y = 73.
What is the meaning of this y-value? It means that, in 1960 (when they started counting), the average lifespan of an American woman was 73 years.
The equation for the speed (not the height) 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. With what initial velocity was the ball thrown? What is the meaning of the slope?
What is the slope? It is m = –32. This value tells me that, for every increase by 1 in my input variable t, I get a decrease of 32 in my output variable v.
What is the meaning of the slope? It means that, every second, the speed decreases by 32 feet per second. Eventually the velocity becomes zero (when the ball reaches its peak), and then becomes negative (when gravity takes over and pulls the ball back down to the ground).
When t = 0, what is the value of v? Looking at the equation, I see that v = 128. The exercise defines v as measuring the velocity of the ball.
What is the meaning of this v-value? It means that, when the ball was released at t = 0 seconds (when they started counting), it was launched upward at 128 feet per second.
Fisherman in the Finger Lakes Region have been recording the dead fish they encounter while fishing in the region. The Department of Environmental Conservation monitors the pollution index for the Finger Lakes Region. The model for the number of fish deaths "y" for a given pollution index "x" is y = 9.607x + 111.958. What is the meaning of the slope? What is the meaning of the y-intercept?
What is the slope? It is m = 9.607. This value tells me that, for every increase by 1 in my input variable x, I get an increase of 9.607 in my output variable y.
What is the meaning of the slope? In means that, for every increase in the pollution index by one unit (say, from a pollution index of 6 to a pollution index of 7), there are nine or ten more fish deaths during the year.
When x = 0, what is the value of y? Looking at the equation, I see that y = 111.958.
What is the meaning of this y-value? It means that, even if the index were zero (that is, even if the water were utterly pure), there would still be about 112 fish deaths a year anyway.
Word problems with linear (straight-line) equations almost always work this way: the slope is the rate of change, and the y-intercept is the starting value.
