For the given equation, first find the x-intercept by setting y = 0: Next, find the y-intercept by setting x = 0:

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1 1. Find the x- and y-intercepts and graph the equation. 5x + 6y = 30 To find the x- and y-intercepts, set one variable equal to 0, and solve for the other variable. To find the x-intercept, set y = 0 and solve for x. The coordinates of the point of the intercept will then be (x, 0). Similarly, to find the y-intercept, set x = 0 and solve for y. The coordinates of the point of the intercept will then be (0, y). For the given equation, first find the x-intercept by setting y = 0: 5x + 6(0) = 30 5x = 30 x = 30/5 x = 6 The x-intercept is then (6, 0). Next, find the y-intercept by setting x = 0: 5(0) + 6y = 30 6y = 30 y = 30 / 6 y = 5 The y-intercept is then (0, 5). To graph the line, plot the two intercepts, (0, 5) and (6, 0), and connect them with a line.

2 2. Find the slope of the line. (See original file for the graph of the line.) To find the slope of the line, you need the coordinates of any two points on the line. The x- and y- intercepts are convenient points, if the coordinates can be determined. In the example, they take one of the intercepts to have coordinates of (0, 4.5). Without any indication of the location of the point halfway between 4 and 5 on the y-axis, such as a tick mark that divides the interval in two equal segments, it is dangerous to interpolate values like this. Instead, try to find two points that are definitely on the line. Using the x-intercept is fine in this case, as it has coordinates that are both whole numbers: (9, 0) For the other point, I would use (3, 3) rather than the y-intercept point, as I cannot be certain from the graph that the coordinates of the y-intercept are exactly (0, 4.5). I am more certain about the point (3, 3) being on the line. The slope then is the difference in the y-coordinates of the two points, divided by the difference in the x-coordinates of the two points. The formula for the slope is: slope = y 2 y 1 x 2 x 1 where x 1 and y 1 are the coordinates of one point, and x 2 and y 2 are the coordinates of the other point. It does not matter which point you designate as point 1 and which you designate as point 2, as long as you are consistent with both the change in y calculation, and the change in x calculation. If you reverse one, then you will end up with the wrong sign on the slope. For the points selected here, (9, 0) and (3, 3), the slope is: slope = = 3 6 = 1 2

3 3. Find the slope of the line through the pair of points (-8, 6) and (3, -7) Use the slope formula from the previous problem: slope = ( 8) = = As mentioned in the previous problem, it doesn t matter if you switch the two points around. Here is the slope calculation using the points in the other order: slope = 6 ( 7) 8 3 = = = How is the graph of a line with a slope equal to 0 situated in a rectangular coordinate system? The equation of a line has the form y = mx + b, where the slope is m. If the slope is zero, then the equation of the line becomes: y = (0)x + b y = b For a line with the equation y = b, the y-coordinate of every point on the line is going to be equal to b, no matter what the x-coordinate is. This line will be depicted as a horizontal line, which will cross the y-axis at a value of y = b. Note: Horizontal lines have slope equal to zero, while vertical lines have an undefined slope. There is a difference. In the case of a vertical line, the change in the x-values is 0. Since the change in x-values goes in the denominator of the slope calculation, the slope ends up being undefined because we cannot divide by zero.

4 5. Find the equation of the line, and write it in (a) slope-intercept form if possible, and (b) standard form. Through (2, -15) and parallel to 2x 9y = 76 First, we need to know the slope of the given line. When the line is given in standard form, ax + by = c, the slope of the line is equal to a/b. In this case, a = 2 and b = -9. The slope is then (2) / (-9) = 2/9. Alternatively, you can rearrange the equation into y = mx + b form and read the slope, m, directly from that equation. Like this: 2x 9y = 76 2x 9y + 9y = y 2x = 9y x 76 = 9y x 76 = 9y 9y = 2x 76 y = (2/9)x (76/9) The slope here is also 2/9, as was found before. Clearly, there is more work involved in this method. Now that we have the slope of this line, we can determine the slope of the other line. Since parallel lines have the same slope, a line parallel to the given line will also have a slope of 2/9. Using the slope-intercept form, we can then write the equation of the new line as: y = 2 9 x + b To determine the value of b, use the given point (2, -15) in place of x and y, and solve for b:

5 15 = 2 ( 9 2) + b 15 = b b = b = 15( 9) b = b = Substituting this value for b, the equation of the line in slope-intercept form is: y = 2 9 x To put this into standard form, first multiply everything by 9 to eliminate the fractions: 2 ( 9)y = ( 9) 9 x ( 9) y = 2x 139 Now rearrange the terms so that the x-term and the y-term are on the same side of the equation, and the coefficient of the x-term is positive: 9y 9y = 2x 9y = 2x 9y = 2x 9y = 2x 9y 2x 9y = 139

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7 6. Find f(-1) when f(x) = -x 2 + 5x 3 To calculate f(-1), substitute -1 in place of each x in the equation: f(x) = -x 2 + 5x 3 f(-1) = -(-1) 2 + 5(-1) 3 f(-1) = -(1) 5 3 f(-1) = f(-1) = Paul invested twice as much money in an account paying 6% interest than he did in an account paying 4% interest. If the total paid was $560, how much did he invest in each? The example showed you how to do this using two variables, and also using just one variable. While the one-variable solution is less work, it requires a higher level of proficiency. I would suggest that you stick to the two-variable method until you are more comfortable with these types of problems. The first sentence of the problem tells us that Paul has invested in two accounts, and that he has invested twice as much in the account that pays 6% as he has in the other account. Let x represent the amount that he invested in the account paying 6%. Let y represent the amount that he invested in the account paying 4%. Since we know that he invested twice as much in the 6% account as he did in the 4% account, we know that x is twice as large as y. We can write this as an equation: x = 2y The amount of interest that is paid in the 6% account will be 0.06x, where x is the amount invested in that account. Similarly, the amount of interest that is paid in the 4% account will be 0.04y, where y is the amount invested in that account. The total interest paid will be the sum of these two quantities:

8 total interest = 0.06x y We also know that the total interest is $560. We can then write a second equation: 0.06x y = 560 Since we know that x = 2y, we can substitute 2y in place of x, and solve for y. Once we know the value of y, we can use the equation x = 2y to determine the value of x. Making the substitution gives: 0.06(2y) y = y y = y = 560 y = 560 / 0.16 y = 3500 The value of x is then: x = 2y x = 2(3500) x = 7000 Solution: Paul invested $7,000 in the account paying 6%, and $3,500 in the account paying 4%.

9 8. Ron and Kathy are ticket-sellers at their class play. Ron is selling student tickets for $3.00 each, and Kathy is selling adult tickets for $5.50 each. Their total income for 12 tickets was $48.50, how many tickets did Ron sell. There are two unknown quantities here: the number of tickets that Ron sold, and the number of tickets that Kathy sold. Let x represent the number that Ron sold. Let y represent the number that Kathy sold. The total number of tickets sold is then x + y. Since we know that they sold a total of 12 tickets, we can write an equation for this: x + y = 12 Since each ticket that Ron sells costs $3, the total income from the sale of Ron s tickets is 3x. Similarly, since each ticket that Kathy sells costs $5.50, the total income from the sale of Kathy s tickets is 5.50y. The total from both Ron and Kathy s ticket sales is then 3x y. We know that the total income is equal to $48.50, so we can write another equation: 3x y = Since we ultimately want to know how many tickets Ron sold (which we have designated as x ), it is desirable to eliminate y from the two equations. To do this, solve the first equation for y, and then substitute it in place of y in the second equation. Solving the first equation, x + y = 12, for y gives: y = 12 x Substituting this in place of y in the second equation gives: 3x (12 x) = x (12) 5.50x = 48.50

10 3x x = x + 66 = x = x = x = / x = 7 Ron sold 7 tickets. 9. During the Junior Hockey League season, the Sharks played 54 games. Together, their wins and losses totaled 51. They tied 17 fewer games than they lost. How many games did they lose that season? Let x represent the number of games that the team lost. Since the total number of wins and losses is 51 we can write: Wins + Losses = 51 Wins + x = 51 Wins = 51 x Since the number of ties is 17 less than the number of losses, the number of ties is: Ties = x 17 There are only three outcomes of a game: a win, a loss, or a tie. And since only one can occur for each game, we have: Wins + Losses + Ties = Games Played Substituting the expressions for wins, losses, and ties gives: (51 x) + x + (x 17) = x + x + x 17 = x 17 = 54

11 x = x = 20 The team had 20 losses. 10. f(x) = 6x 3 4x 2 x + 42; f(-2) Substitute -2 in place of each x in f(x) and calculate: f(x) = 6x 3 4x 2 x + 42 f(-2) = 6(-2) 3 4(-2) 2 (-2) + 42 (-2) 3 is equal to 8, (-2) 2 is equal to 4, and (-2) is equivalent to +2 Making those changes gives: f(-2) = 6(-8) 4(4) f(-2) = f(-2) = Find the product: (-5x 1) 2 (-5x 1) 2 = (-5x 1)(-5x 1) Using the FOIL method: First: (-5x)(-5x) = 25x 2 Outer: (-5x)(-1) = 5x Inner: (-1)(-5x) = 5x Last: (-1)(-1) = 1 Adding the four results together gives: (-5x 1) 2 = 25x 2 + 5x + 5x + 1 (-5x 1) 2 = 25x x + 1

12 12. (a 10)(a + 10) Again, using the FOIL Method gives: First: (a)(a) = a 2 Outer: (a)(10) = 10a Inner: (-10)(a) = -10a Last: (-10)(10) = -100 Adding the four results together gives: (a 10)(a + 10) = a a 10a 100 (a 10)(a + 10) = a If you can recognize the pattern of factors as (a + b)(a b), then you can go directly to the answer since (a + b)(a b) = a 2 b 2. This is known as the difference of two squares. 13. (x 5)(3x 10) Using the FOIL Method gives: First: (x)(3x) = 3x 2 Outer: (x)(-10) = -10x Inner: (-5)(3x) = -15x Last: (-5)(-10) = 50 Adding the four results together gives: (x 5)(3x 10) = 3x 2 10x 15x + 50 (x 5)(3x 10) = 3x 2 25x + 50

13 14. 9x 2 + 6x 8 3x 2 This process is similar to the long division that you probably learned in grade school. It s a little more complicated, but the principle is the same. First, determine how many times 3x will divide into 9x 2. Or, to put it another way, calculate 9x2 3x. The result is 3x. Write this result above the division bar. I prefer to write this result directly over the 9x 2 term, but the example that you were given seems to want to place it over the 6x term. I ll do it the same way so as not to add any confusion. 3x Next, multiply 3x 2 by 3x. The result is 9x 2 6x. Write this result in a new line under the expression 9x 2 + 6x 8: 3x 9x 2 6x Next, subtract 9x 2 6x from 9x 2 + 6x 8, and write the result at the bottom of the problem, like this: 3x 9x 2 6x 0x 2 +12x Next, bring the -8 term down and write it next to the 12x term in the bottom line:

14 3x 9x 2 6x 12x 8 Next, divide 3x into 12x: 12x 3x = 4 Write this result above the division bar, to the right of the first 3x: 3x + 4 9x 2 6x 12x 8 Next, multiply 3x 2 by 4. The result is 12x 8. Write this result on a new line at the bottom of the problem: 3x + 4 9x 2 6x 12x 8 12x 8 Subtract the last line, 12x 8, from the line above it. The result is 0: 3x + 4 9x 2 6x 12x 8 12x 8 Since the result is 0, there is no remainder. The solution is 3x