ProblemA small business has a debt of $500,000$500,000 incurred from start-up costs. It predicts that it can pay off the debt at a rate of $85,000$85,000 per year according to the following equation governing years in business (x𝑥) and debt measured in thousands of dollars (y𝑦).y=−85x+500𝑦=−85𝑥+500Graph the above equation and use your graph to predict when the debt will be fully paid. SolutionAgain it is clear from the function we wrote that this scenario represents a linear function. First, we start with creating a table of values:x𝑥y𝑦0050050011Answer 1 Question 32233033033Answer 2 Question 344160160Then we plot our points and draw the line that goes through them:Notice the scale we’ve chosen here. There’s no need to include any points above y=500𝑦=500, but it’s still wise to allow a little extra. The y𝑦-intercept of this function would be (0,(0, Answer 3 Question 3 )), which represents the amount of debt they began with.The slope, or overall pattern of a linear graph, can be found by using two points on the line. We can automatically notice that the slope will be negative, since the y𝑦-values get smaller left to right. It is always best to choose points on a graph that land close to the intervals marked, for simpler math and higher accuracy. For this reason, we’ll choose two points from our table above: (0,500)(0,500) and (3,245)(3,245). We’ll plug these in the slope formula:m=y2−y1x2−x1=245−5003−0=𝑚=𝑦2−𝑦1𝑥2−𝑥1=245−5003−0=Answer 4 Question 3 == Answer 5 Question 333Had we used our graph to find the slope, it would’ve been hard because our points are not on the intervals. The point for 33 years LOOKS like it is on 250250, but when we double check our chart, it is actually at 245245. Take a look at the scale on the y𝑦-axis. It is increasing in increments of $50$50 which makes it very difficult to determine the exact values of each point. What do you think a better scale may have been for the y𝑦-axis? [click to open]If we look at our table of values, it looks like if we used a scale of five, several of our points would now land on the intervals, making it much easier to identify the values. This would make our graph larger, but would make interpreting it much easier.While this numerical value of the slope doesn’t simplify further like we’ve previously seen, it still can give us valuable information about the pattern occurring. Remember that the slope formula represents the change in y𝑦 over the change in x𝑥, so for our problem, it represents the change in the amount of debt over the change in the number of years in business. So, we can say that for every three years in business, we can expect the debt to be reduced (go down) by 250,000250,000. (Remember, our units were in thousands of dollars.)The domain and range of this function are both Answer 6 Question 3 or x≥0𝑥≥0 and y≥0𝑦≥0. [click to open]Our output value/dependent variable is debt. Would it make sense for the amount of debt to ever go below zero? The input value/independent variable is the number of years in business. Would a company ever be in business for less than zero years? Would that make sense?Next we need to determine how many years it takes the debt to reach zero, or in other words, what x𝑥−value will make the y𝑦−value equal zero. We know it’s greater than four (since at x=4𝑥=4 the y𝑦−value is still positive), so we need an x𝑥−scale that goes well past x=4𝑥=4. Here we’ve chosen to show the x𝑥−values from 00 to 1212, though there are many other places we could have chosen to stop. We’ve also chosen a scale of 11 for the x𝑥-axis because when we count by years, we usually count by ones.To read the time that the debt is paid off, we simply read the point where the line hits y=0𝑦=0 (the x𝑥−axis). It looks as if the line hits pretty close to x=𝑥= Answer 7 Question 3. (Type a whole number.)So the debt will definitely be paid off in Answer 8 Question 3 years.CheckQuestion 3

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ProblemA small business has a debt of $500,000$ 500,000 incurred from start-up costs. It predicts that it can pay off the debt at a rate of $85,000$ 85,000 per year according to the following equation governing years in business (x𝑥) and debt measured in thousands of dollars (y𝑦).y=−85x+500𝑦=−85𝑥+500Graph the above equation and use your graph to predict when the debt will be fully paid. SolutionAgain it is clear from the function we wrote that this scenario represents a linear function. First, we start with creating a table of values:x𝑥y𝑦0050050011Answer 1 Question 32233033033Answer 2 Question 344160160Then we plot our points and draw the line that goes through them:Notice the scale we’ve chosen here. There’s no need to include any points above y=500𝑦=500, but it’s still wise to allow a little extra. The y𝑦-intercept of this function would be (0,(0, Answer 3 Question 3 )), which represents the amount of debt they began with.The slope, or overall pattern of a linear graph, can be found by using two points on the line. We can automatically notice that the slope will be negative, since the y𝑦-values get smaller left to right. It is always best to choose points on a graph that land close to the intervals marked, for simpler math and higher accuracy. For this reason, we’ll choose two points from our table above: (0,500)(0,500) and (3,245)(3,245). We’ll plug these in the slope formula:m=y2−y1x2−x1=245−5003−0=𝑚=𝑦2−𝑦1𝑥2−𝑥1=245−5003−0=Answer 4 Question 3 == Answer 5 Question 333Had we used our graph to find the slope, it would’ve been hard because our points are not on the intervals. The point for 33 years LOOKS like it is on 250250, but when we double check our chart, it is actually at 245245. Take a look at the scale on the y𝑦-axis. It is increasing in increments of $50$ 50 which makes it very difficult to determine the exact values of each point. What do you think a better scale may have been for the y𝑦-axis? [click to open]If we look at our table of values, it looks like if we used a scale of five, several of our points would now land on the intervals, making it much easier to identify the values. This would make our graph larger, but would make interpreting it much easier.While this numerical value of the slope doesn’t simplify further like we’ve previously seen, it still can give us valuable information about the pattern occurring. Remember that the slope formula represents the change in y𝑦 over the change in x𝑥, so for our problem, it represents the change in the amount of debt over the change in the number of years in business. So, we can say that for every three years in business, we can expect the debt to be reduced (go down) by 250,000250,000. (Remember, our units were in thousands of dollars.)The domain and range of this function are both Answer 6 Question 3 or x≥0𝑥≥0 and y≥0𝑦≥0. [click to open]Our output value/dependent variable is debt. Would it make sense for the amount of debt to ever go below zero? The input value/independent variable is the number of years in business. Would a company ever be in business for less than zero years? Would that make sense?Next we need to determine how many years it takes the debt to reach zero, or in other words, what x𝑥−value will make the y𝑦−value equal zero. We know it’s greater than four (since at x=4𝑥=4 the y𝑦−value is still positive), so we need an x𝑥−scale that goes well past x=4𝑥=4. Here we’ve chosen to show the x𝑥−values from 00 to 1212, though there are many other places we could have chosen to stop. We’ve also chosen a scale of 11 for the x𝑥-axis because when we count by years, we usually count by ones.To read the time that the debt is paid off, we simply read the point where the line hits y=0𝑦=0 (the x𝑥−axis). It looks as if the line hits pretty close to x=𝑥= Answer 7 Question 3. (Type a whole number.)So the debt will definitely be paid off in Answer 8 Question 3 years.CheckQuestion 3

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