Review for module 6

1. Select the system of equations that corresponds to the given graph. 1. Look at the graph to see where the line crosses the y-axis. This is your y-intercept or ‘b’ value to be used In the slope intercept form y = mx+b Both lines cross at 3. So for both equations the y-int is at 3, or b=3. Find the slope of each line. This is your ‘m’ value. You may choose any two points on the line and Calculate slope using m = (y2-y1)/(x2-x1) where (x1,y1) is your first point and (x2,y2) is your second point. Or you can count grid lines on the graph as you move from the intersection of one grid line to another. 1st line: move down 1 unit and right 2 units. So slope is -1/2. m=-1/2 2nd line: move down 2 units, right 1 unit. So slope is -2/1. m= -2 Put your ‘b’ value and your ‘m’ value into slope intercept form y=mx+b So for the 1st line you have y=(-1/2)x +3 and for the 2nd line you have y=-2x+3

2. Is the graph of the following system of equations parallel lines, intersecting lines, or overlapping lines? -2x + y = 3 -4x + 2y = 6 Solve each equation for ‘y’ so it is in slope-intercept form y = mx+b If the equations are identical – have the same slope ‘m’ and y-intercept ‘b’ values, then they are overlapping lines. If the equations have the same slope ‘m’ value but have different y-intercept ‘b’ values, they are parallel. If neither of these are true, then the lines will intersect. -2x + y =3 add 2x to both sides -4x + 2y =6 add 4x to both sides -2x + y + 2x = 3+ 2x -4x + 2y + 4x = 6 + 4x y = 2x + 3 2y = 4x + 6 divide every term by 2 2y/2 = 4x/2 + 6/2 y = 2x +3 These are the same equation.

3. For the following system, if you isolated y in the first equation to use the Substitution Method, what expression would you substitute into the second equation? 2x + y = 8 Solve for y by subtracting 2x from both sides. -x – y = -5 2x + y – 2x = 8-2x y = 8 -2x or can write as y = -2x+8 If asked how to solve this is what you would do next: Now substitute that into second equation (remember to use parenthesis): -x – (-2x + 8) = -5 Use distributive property to distribute the negative sign -x +2x – 8 = -5 Then combine like terms x-8 = -5 Add 8 to both sides x= 3

4. Look at the solution to the system of equations below. Was a mistake made? If so what was it?3x + y = 6x – 2y = 2y = 6 – 3xx – 2(6 – 3x) = 2x – 12 + 6x = 27x – 12 = 2 + 12 +12 x = 23(2) + y = 6 6 + y = 6–6 –6 y = 0 All the math here checks out. This problem was solved correctly.

5. There are a total of 90 boys and girls who play sports. If the number of boys is 10 more than three times the number of girls, how many girls play sports at this high school? Define variables for your unknowns. Write two separate equations. One for each sentence. Let b= number of boys and g= number of girls b+g = 90 and b=3g+10 Solve one of the equations in terms of what you are trying to solve for. Solve the first equation for b by subtracting g from both sides: b = 90-g Use substitution method to solve. Remember to use parenthesis: Instead of b = 3g+10 we now have 90-g = 3g+10 then add g to both sides 90-g+g = 3g+10+g combine like terms 90 = 4g+10 Subtract 10 from both sides 90-10 = 4g+10-10 80 = 4g divide both sides by 4 80/4 = 4g/4 20 =g Therefore 20 girls played sports.

6. Solve the following system of equationsy = x + 45x + y = 16 Since one of the equations is already solved for ‘y’ it is easiest to use substitution method. Always remember to use parenthesis when substituting. Substitute the first equation into the second equation. 5x + (x+4) = 16 combine like terms 6x+4 = 16 subtract 4 from both sides 6x+4-4 = 16-4 6x = 12 divide both sides by 6 6x/6 = 12/6 x= 2 Remember to go back and substitute your solution into any of the equations and solve for the other variable. Remember to write your final answer as an ordered pair. y= x+ 4 becomes y = (2) + 4 or y = 6 Final answer is (2,6)

7. A total of 124 men and women tried out for the lead singer position. If twice the number of men less three times the number of women is equal to 28, how many women auditioned? Define variables. Write two separate equations, one for each sentence. Let m=number of men and w=number of women 124 = m+w and 2m-3w=28 Solve one of the equations in terms of what you are trying to solve for. Solve the first equation for m by subtracting w from both sides. 124-w = m+w-w 124-w = m Use substitution method to solve. Remember to use parenthesis. 2m-3w=28 becomes 2(124-w) -3w = 28 Use distributive property to get 248 – 2w -3w = 28 combine like terms 248 -5w = 28 then subtract 248 from both sides 248-5w-248 = 28-248 -5w = -220 divide both sides by -5 -5w/-5 = -220/-5 w = 44 so 44 women auditioned

8. Write the equation you would use to solve the following word problem:Sasha's mom decided to get her a cell phone for her birthday. Cell Plus has a plan that costs $22.95 per month plus an additional $0.08 per minute. Cell Best has a plan that costs $27.95 per month plus an additional $0.06 per minute. How many minutes can Sasha talk and have the same cost each month? Define a variable for your unknown. Since the problem asks when the costs will be the same, set the two equations equal to each other. Let m = number of minutes Cell Plus plan: 22.95+ 0.08m Cell Best plan: 27.95 + 0.06m Set equal to each other since it says cost each month is the same 22.95 + 0.08m = 27.95 + 0.06m then subtract 22.95 from both sides 22.95 + 0.08m -22.95 = 27.95 + 0.06 – 22.95 0.08m = 5 + 0.06m then subtract 0.06m from both sides 0.08m – 0.06m = 5 + 0.06m -0.06m 0.02m = 5 divide both sides by 0.02 0.02m/0.02 = 5/0.02 m=250

9. There were 150 people at the Junior-Senior dance. Junior tickets were $2.00 each and Senior tickets were $3.50 each. The total receipts for the dance were $405. How many Juniors bought tickets? Define variables. Write two separate equations, one for each sentence. Let j=number of juniors and Let s= number of seniors j+s =150 and 2j + 3.5s = 405 Solve one of the equations in terms of what you are trying to solve for. Solve the first equation for s by subtracting j from both sides j+s-j = 150 –j s=150-j Use substitution method to solve. Remember to use parenthesis. 2j+3.5s = 405 becomes 2j+3.5(150-j) = 405 use distributive property 2j + 525 – 3.5j = 405 combine like terms -1.5j + 525 = 405 then subtract 525 from both sides -1.5j+525-525 = 405-525 -1.5j = -120 divide both sides by -1.5 -1.5j/-1.5 = -120/-1.5 j= 80 so 80 juniors bought tickets

10. Write a set of equations that would be used to solve this word problem:An airplane flew 4.5 hours with a 45 mph head wind. The return trip with a tail wind of the same speed took 2.5 hours. Find the speed of the plane in still air. Since this problem talks about distance, rate and time, use the equation d=rt Where the rate is the combined plane speed and wind speed. If it is a head wind, that means it is pushing against the plane slowing it down. That is why it takes more time. So r= plane speed – wind speed If it is a tail wind, that means it is pushing with the plane speeding it up. That is why it takes less time. So r= plane speed + wind speed. Let p= speed of the plane Distance(with head wind) = (p-45)(4.5) and Distance(with tail wind) = (p+45)(2.5) If asked to solve, since the distance is the same both ways, set the equations equal to each other. (p-45)(4.5) = (p+45)(2.5) use distributive property 4.5p-202.5 = 2.5p + 112.5 then subtract 2.5p from both sides 4.5p – 202.5 -2.5p = 2.5p + 112.5 -2.5p 2p – 202.5 = 112.5 then add 202.5 to both sides 2p = 315 then divide both sides by 2 2p/2 = 315/2 p= 157.5 mph

11. Select the system of inequalities that corresponds to the given graph Look at the graph to see where the line crosses the y-axis. This is your y-intercept or ‘b’ value to be used in the slope intercept form y = mx+b 1st line crosses at 2 so b=2. 2nd line crosses at -6 so b= -6 Find the slope of each line. This is your ‘m’ value. You may choose any two points on the line and calculate slope using m = (y2-y1)/(x2-x1) where (x1,y1) is your first point and (x2,y2) is your second point. Or you can count grid lines on the graph as you move from the intersection of one grid line to another. 1st line: move down 2 units and right 1 unit so slope=-2/1 or m = -2 2nd line: move up 1 unit and right 1 unit so slope=1/1 or m=1 Put your ‘b’ value and your ‘m’ value into slope intercept form y=mx+b 1st line: y = -2x+2 and 2nd line: y = x-6 4. Look at the graph to determine which inequality symbol to use. If the line is dashed it has to be < or >. If it is a solid line then it is ≤ or ≥. Then look at which side of the line the shading is on. If it is shaded up above the line choose the symbol that had > as part of it. If it is shaded down below the line choose the symbol that had < as part of it. 1st inequality: y > -2x+2 and 2nd inequality: y > x-6

12. For the following system of equations, write your own real world scenario that describes what is happening. 2x + y = 93x + 4y = 26this could be the cost of children tickets(x) and adult tickets(y) Solve the system and explain what the results mean according to your scenario. 2x + y = 9 3x + 4y = 26 Solvethe first equation for ‘y’. Subtract 2x from both sides to get y = 9-2x Substitute that into the second equation. Remember to use parenthesis. 3x+4y=26 becomes 3x+4(9-2x) = 26 use distributive property 3x + 36 – 8x = 26 combine like terms -5x + 36 = 26 subtract 36 from both sides -5x+36-36 = 26-36 -5x = -10 divide both sides by -5 -5x/-5 = -10/-5 x = 2 Remember to go back and substitute your solution into any of the equations and solve for the other variable. Remember to write your final answer as an ordered pair. Now y=9-2x becomes y = 9-2(2) or y = 9-4 or y=5 Solution is (2,5) or the price of a child ticket is $2.00 and price of an adult ticket is $5.00

13. Your piggy bank has a total of 47 coins in it; some are dimes and some are nickels. If you have a total of $3.95, how many nickels and how many dimes do you have? Define variables for your unknowns. Write two separate equations, one for each sentence. The first one will be about the number of coins. The second will be about the dollar value worth of the coins. Let d= number of dimes. Let n= number of nickels d+n = 47 and since a dime is worth $0.10 and a nickel is worth $0.05 you have 0.10d+0.05n=3.95 Solve the first equation for one of the variables. Then substitute that into the second equation. Remember to use parenthesis. Solve first equation for n by subtracting d from both sides: n=47-d So 0.10d+0.05n=3.95 becomes 0.10d + 0.05(47-d)= 3.95 use distributive property to get 0.10d +2.35 – 0.05d = 3.95 combine like terms 0.05d +2.35 = 3.95 then subtract 2.35 from both sides 0.05d + 2.35 -2.35 = 3.95-2.35 0.05d = 1.60 divide both sides by 0.05 0.05d/0.05 = 1.60/0.05 d = 32 Remember to go back and substitute your solution into any of the equations and solve for the other variable. n=47-d becomes n=47-32 so n=15. There were 32 dimes and 15 nickels.

14. Solve the following system of equations:x – y = 10x + y = 8 Since the coefficients of the ‘y’ variables in both equations are 1 and -1, we can solve by using elimination method. Add the two equations together to solve. x – y = 10 + x + y =8 ------------------------ 2x + 0 = 18 or 2x =18 then divide both sides by 2 2x/2 = 18/2 x = 9 Remember to go back and substitute your solution into any of the equations and solve for the other variable. Remember to write your final answer as an ordered pair. Now x + y =8 becomes (9)+y =8 subtract 9 from both sides 9 + y -9 = 8-9 y = -1 Solution is (9,-1)