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Question 71195: The graphics calculator display shows a table of x- and y- values for a linear equation y=mx+b. Determine the slope and y-intercept.
x ------ y1
-3 ------ 4.5
-2 ------ 4
-1 ------ 3.5
0 ------ 3
1 ------ 2.5
2 ------ 2
3 ------ 1.5
x=-3
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! This problem can be done by inspection. Think about the equation y = mx + b. The slope is m and the b is the value on the y-axis where the graph crosses the y-axis. This form of equation is called, for obvious reasons, the slope-intercept form.
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Ask yourself, "What is the value of x for any value of y on the y-axis?" Think about a graph. If you mark a spot on the y-axis, the corresponding value of x is zero, isn't it? And working in reverse, you can say that when x is zero, the corresponding value of y is on the y-axis. As an extension of this, when x = 0, the corresponding value for y has to be the value of "b" in slope-intercept equation. Look at the table of values you were given. When x = 0, the corresponding value of y is +3. Therefore, we can say b is +3. Substituting this information into the slope-intercept equation, the equation becomes y = mx + 3. We've gotten a step closer to the answer.
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Now all we need is the slope of the graph and we will be able to finish the equation. So let's begin by thinking about the definition of slope. Slope is defined as the change in y for a corresponding change in x. Well, let's pick a convenient value for the change in x. Let's pick a change in x of 1 unit. If we do, then in finding the slope we will be dividing by 1, an easy division to do in your head. Let's go to the table and pick any convenient value of x. Then we will increase it by 1 and find out what the corresponding change in y is.
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How about if we start with x = 0 and let x increase to x = 1. For that same interval y goes from +3 to y = 2.5, a change of - .5. So -.5 is the slope because slope = -.5/1 ... the change in y divided by the corresponding change in x.
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Now that we have the slope, we can write the equation. It is:
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y = (-.5)x + 3
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You can check this equation by selecting a value for x from the table, plugging it into the right side of the equation, and seeing if the solution for y doesn't correspond to the value for y in the table. For example, let's select x = 2 from the table. Plugging that into the equation we get:
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y = (-.5)*2 + 3 = -1 + 3 = +2
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So the equation tells us that if x = 2 then y = +2. And that's what the table says too. You can do the same things for other values for x and see if the corresponding values of y that you calculate agree with the table.
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Hope this helps you understand the problem a little better and gives you some insight into how the slope intercept form of a linear equation works.
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