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Equations/183278: how do you solve 3w-4=w+8 1 solutions Answer 137613 by jim_thompson5910(28476)   on 2009-02-22 21:29:25 (Show Source): You can put this solution on YOUR website! Start with the given equation. Add to both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is

Equations/183231: This question is from textbook integrated algebra 1 15x-3(3x+4)=6 1 solutions Answer 137611 by jim_thompson5910(28476)   on 2009-02-22 21:17:09 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Combine like terms on the left side. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is

Expressions-with-variables/183267: How do you factor (x^2+2xy+y^2)-4w^2? 1 solutions Answer 137610 by jim_thompson5910(28476)   on 2009-02-22 21:12:01 (Show Source): You can put this solution on YOUR website! Start with the given expression. First, we need to factor Looking at we can see that the first term is and the last term is where the coefficients are 1 and 1 respectively. Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient 2? Let's list all of the factors of 1: Factors of 1: 1 -1 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to 1 1*1 (-1)*(-1) note: remember two negative numbers multiplied together make a positive number Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2 First Number Second Number Sum 1 1 1+1=2 -1 -1 -1+(-1)=-2 From this list we can see that 1 and 1 add up to 2 and multiply to 1 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) note: is equivalent to since the term occurs twice. So also factors to So factors to --------------------------------------------------------------- So the original expression becomes Rewrite as Notice how we now have a difference of squares. Remember, the difference of squares formula is In this case, and Now plug them in the formula to get: ================================================== Answer: So this means that factors to Equivalently, this means that factors to

Linear-systems/183269: Use elimination method:tnx 5x-3y=32 4x+3y=4 1 solutions Answer 137609 by jim_thompson5910(28476)   on 2009-02-22 21:04:56 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)

Linear-systems/183271: Use elimination method 4x-y=8 2x+y=4 1 solutions Answer 137605 by jim_thompson5910(28476)   on 2009-02-22 20:53:57 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)

Linear-systems/183268: Use elimination method in solving this:tnx 4x-y=8 2x+y=4 1 solutions Answer 137604 by jim_thompson5910(28476)   on 2009-02-22 20:53:15 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)

Linear-systems/183262: Use elimination method in solving this;tnx x+3y=5 2x+y=5 1 solutions Answer 137602 by jim_thompson5910(28476)   on 2009-02-22 20:22:00 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Multiply the both sides of the first equation by -2. Distribute and multiply. So we have the new system of equations: Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)

Angles/183256: This question is from textbook Blitzer- Introductory Algebra The measure of the angle's supplement is 40 degrees more than 3 times that of its complement. Find the measure of the angle described. I know the answer is 65 but not sure how to find the answer. This is what I came up with but it doesn't work out correctly. x=3(180-x)+40 I'd sure appreciate your help. I'm in my 40's and going to college for the first time. Math isn't coming easy but I'm really trying my best. Thanks 1 solutions Answer 137601 by jim_thompson5910(28476)   on 2009-02-22 20:20:59 (Show Source): You can put this solution on YOUR website! Let x=unknown angle So the supplement is 180-x and the compliment is 90-x This means that "The measure of the angle's supplement is 40 degrees more than 3 times that of its complement" translates to Start with the given equation. Distribute. Combine like terms on the right side. Subtract from both sides. Add to both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is which means that the unknown angle is 65 degrees.

Complex_Numbers/183246: 25v^2-9 1 solutions Answer 137586 by jim_thompson5910(28476)   on 2009-02-22 19:35:41 (Show Source): You can put this solution on YOUR website! I'm assuming that you want to factor this. Start with the given expression. Rewrite as . Rewrite as . Notice how we have a difference of squares where in this case and . So let's use the difference of squares formula to factor the expression: Start with the difference of squares formula. Plug in and . So this shows us that factors to . In other words .

Distributive-associative-commutative-properties/183247: 5y^2-28y-12 1 solutions Answer 137585 by jim_thompson5910(28476)   on 2009-02-22 19:34:28 (Show Source): You can put this solution on YOUR website! I'm assuming that you want to factor this. Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,2,3,4,5,6,10,12,15,20,30,60 -1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*(-60) 2*(-30) 3*(-20) 4*(-15) 5*(-12) 6*(-10) (-1)*(60) (-2)*(30) (-3)*(20) (-4)*(15) (-5)*(12) (-6)*(10) Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 -60 1+(-60)=-59 2 -30 2+(-30)=-28 3 -20 3+(-20)=-17 4 -15 4+(-15)=-11 5 -12 5+(-12)=-7 6 -10 6+(-10)=-4 -1 60 -1+60=59 -2 30 -2+30=28 -3 20 -3+20=17 -4 15 -4+15=11 -5 12 -5+12=7 -6 10 -6+10=4 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term --------------------------------------------- Answer: So factors to . In other words, . Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Distributive-associative-commutative-properties/183250: w^2+14w+49 1 solutions Answer 137584 by jim_thompson5910(28476)   on 2009-02-22 19:33:16 (Show Source): You can put this solution on YOUR website! I'm assuming that you want to factor this. Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,7,49 -1,-7,-49 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*49 7*7 (-1)*(-49) (-7)*(-7) Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 49 1+49=50 7 7 7+7=14 -1 -49 -1+(-49)=-50 -7 -7 -7+(-7)=-14 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term Condense the factors --------------------------------------------- Answer: So factors to . In other words, Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Complex_Numbers/183249: 3x^2-4x-4 1 solutions Answer 137583 by jim_thompson5910(28476)   on 2009-02-22 19:31:49 (Show Source): You can put this solution on YOUR website! I'm assuming that you want to factor this. Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,2,3,4,6,12 -1,-2,-3,-4,-6,-12 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*(-12) 2*(-6) 3*(-4) (-1)*(12) (-2)*(6) (-3)*(4) Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 -12 1+(-12)=-11 2 -6 2+(-6)=-4 3 -4 3+(-4)=-1 -1 12 -1+12=11 -2 6 -2+6=4 -3 4 -3+4=1 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term --------------------------------------------- Answer: So factors to . Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Complex_Numbers/183248: z^2-10z-24 1 solutions Answer 137582 by jim_thompson5910(28476)   on 2009-02-22 19:31:07 (Show Source): You can put this solution on YOUR website! I'm assuming that you want to factor this. Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,2,3,4,6,8,12,24 -1,-2,-3,-4,-6,-8,-12,-24 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*(-24) 2*(-12) 3*(-8) 4*(-6) (-1)*(24) (-2)*(12) (-3)*(8) (-4)*(6) Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 -24 1+(-24)=-23 2 -12 2+(-12)=-10 3 -8 3+(-8)=-5 4 -6 4+(-6)=-2 -1 24 -1+24=23 -2 12 -2+12=10 -3 8 -3+8=5 -4 6 -4+6=2 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term --------------------------------------------- Answer: So factors to . Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Linear-systems/183235: 4c+3d=-2 8c-2d=12 1 solutions Answer 137580 by jim_thompson5910(28476)   on 2009-02-22 18:51:14 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Multiply the both sides of the first equation by -2. Distribute and multiply. So we have the new system of equations: Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So our answers are and . This means that the system is consistent and independent.

Rational-functions/183233: This question is from textbook Intermediate algebra 6x^2-11x-10/2x^2-9x+10 1 solutions Answer 137578 by jim_thompson5910(28476)   on 2009-02-22 18:38:17 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get . Factor to get . Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where or

Percentage-and-ratio-word-problems/183221: What is 5/15 as a percentage? 1 solutions Answer 137571 by jim_thompson5910(28476)   on 2009-02-22 17:56:15 (Show Source): You can put this solution on YOUR website! note: the '3's after the decimal repeat forever. Now multiply by 100 to get 33.33% So as a percentage is

Quadratic_Equations/183220: The yellow bus stops at a bus stop every 15 minutes. The blue bus stop at the same bus stop every 20 minutes. If both buses reach the bus stop at 8:30 a.m., what is the next time both the yellow and blue buses will reach the bus stop at the same time? 1 solutions Answer 137569 by jim_thompson5910(28476)   on 2009-02-22 17:53:17 (Show Source): You can put this solution on YOUR website! Since the LCM of 15 and 20 is 60, this means that it will take 60 minutes (1 hour) for the buses to meet at the same time again. So an hour from 8:30 a.m. is 9:30 a.m. So the buses will meet at 9:30 a.m. Here's a table to confirm the answer: Yellow | Blue ----------------- 8:30 | 8:30 8:45 | 8:50 9:00 | 9:10 9:15 | 9:30 9:30 |

Exponents-negative-and-fractional/183223: Exponents: Simplify the expression if possible. Write your answer as a power. (-2x)4th power 1 solutions Answer 137567 by jim_thompson5910(28476)   on 2009-02-22 17:41:38 (Show Source): You can put this solution on YOUR website! So

Exponents/183219: (3z^4 - 8)^2 1 solutions Answer 137566 by jim_thompson5910(28476)   on 2009-02-22 17:39:45 (Show Source): You can put this solution on YOUR website! Start with the given expression. Expand. Remember something like . Now let's FOIL the expression. Remember, when you FOIL an expression, you follow this procedure: Multiply the First terms:. Multiply the Outer terms:. Multiply the Inner terms:. Multiply the Last terms:. --------------------------------------------------- So we have the terms: , , , and Now collect every term listed above to make a single expression. Now combine like terms. So FOILs to . In other words, .

Linear-equations/183218: This problem has more than one part, please HELP ME put it all together.? The price of unleaded regular gasoline varies with the price per barrel of oil on the world market. When oil was selling for $30 per barrel I paid $1.25 per gallon for gasoline. When oil was selling for $140 per barrel, I paid $4.00 for a gallon of gasoline. I have to make a table showing the above data for gasoline and oil prices. Then, I have to construct a graph with oil price per barrel on the horizontal axis and the price per gallon of gasoline on the vertical axis, then plot the two points given and draw a line connecting the two points and extending over the range of $0 to $150 for the price of a barrel of oil, and find the slope of the line, then find the “y” intercept for the line (the point where oil is $0.00 per barrel) and finally write the equation of the line. Assuming that the relationship is linear, I have to calculate the price for a gallon of gasoline when the oil price reaches $39.34 per barrel. How does this compare to the price you are paying today? 1 solutions Answer 137565 by jim_thompson5910(28476)   on 2009-02-22 17:37:53 (Show Source): You can put this solution on YOUR website! Let x=price per barrel of oil and y=price per gallon of gasoline First translation: "When oil was selling for $30 per barrel I paid $1.25 per gallon for gasoline" means that when , . So we have one point (130,1.25) Second translation: "When oil was selling for $140 per barrel, I paid $4.00 for a gallon of gasoline" means that when , . So we have another point (140,4) So here's the table of the two ordered pairs (points) x y 130 1.25 140 4 Now set up the axis (with the proper ranges and labels) Plot the two points Draw a line through the points (in blue) ----------------------------------------------------------------- Now let's find the equation of the line that goes through the points (130,1.25) and (140,4) First let's find the slope of the line through the points (130,1.25) and (140,4) Note: is the first point (130,1.25) and is the second point (140,4) Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Divide So the slope of the line that goes through the points (130,1.25) and (140,4) is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Distribute Multiply Add to both sides. Combine like terms. So the equation that goes through the points (130,1.25) and (140,4) is The equation is now in slope intercept form where the slope is and the y-intercept is Note: Since the slope is , this means that for every dollar increase that the price per barrel of oil experiences, the price per gallon of gas will increase $0.275 Also, the y-intercept is the value when the price of oil is $0 per barrel. So if the price per barrel is $0, then the price of gas is -34.50 dollars. ------------------------------------------------------------------- "Assuming that the relationship is linear, I have to calculate the price for a gallon of gasoline when the oil price reaches $39.34 per barrel" In this case, we want to know "y" when "x" is equal to 39.34. So simply plug in to find "y" Start with the equation we just found Plug in Multiply Subtract So when the price of oil is $39.34 a barrel, the price of gas will be about -23.68 dollars a barrel. Note: the relationship between the price of oil and the price of gas is a little more complex than just a simple linear relationship (since there are more factors involved).

Equations/183211: "find all integers m for which y^2+my+50 can be factored?" 1 solutions Answer 137556 by jim_thompson5910(28476)   on 2009-02-22 16:26:52 (Show Source): You can put this solution on YOUR website! First, multiply the first coefficient 1 and the last term 50 to get 50. Now list the factors of 50 Factors of 50: 1,2,5,10,25,50 -1,-2,-5,-10,-25,-50 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*50 2*25 5*10 (-1)*(-50) (-2)*(-25) (-5)*(-10) Now add up each paired factor (ie 1+50=51, 2+25=27, etc..): First Number Second Number Sum 1 50 1+50=51 2 25 2+25=27 5 10 5+10=15 -1 -50 -1+(-50)=-51 -2 -25 -2+(-25)=-27 -5 -10 -5+(-10)=-15 All of the numbers in the last column are possible values for the value of "m". Remember, you can only factor only if the factors of add to "b" So the possible values for "m" are: 51, 27, 15, -51, -27, -15 Note: all of these values of "m" are straight from that third column. This means that the following quadratics are factorable: , , , , , and

real-numbers/183200: "suppose x^2+a^2x+a^2 factors into (x+a)^2. what is the value of a, that is not zero?" 1 solutions Answer 137554 by jim_thompson5910(28476)   on 2009-02-22 16:07:12 (Show Source): You can put this solution on YOUR website! Since we're supposing that "x^2+a^2x+a^2 factors into (x+a)^2", this means that Start with the given equation. FOIL the right side Subtract from both sides. Note: the terms cancel out. Subtract from both sides. Note: the terms cancel out. So we're left with: Divide both sides by "x". Once again, the "x" terms cancel Simplify Subtract 2a from both sides. Factor the left side Now set each factor equal to zero: or or Now solve for "a" in each case Note: since we want a value of "a" "that is not zero", this means that we'll ignore the value =========================================================== Answer: So the solution is

Quadratic-relations-and-conic-sections/183194: Given that the equation of a circle is x^2+y^2-10x+4y+13=0, find its center and its radius. 1 solutions Answer 137551 by jim_thompson5910(28476)   on 2009-02-22 15:51:34 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract 13 from both sides. Group like terms. Take half of the "x" coefficient -10 to get -5. Square it to get 25. Add this to both sides. Take half of the "y" coefficient 4 to get 2. Square that result to get 4. Add this to both sides. Combine like terms. Factor to get Factor to get Rewrite as Rewrite as So the equation is now in the form (which is a circle) where (h,k) is the center and "r" is the radius We can see that , , and So the center is (5,-2) and the radius is 4 units

Rational-functions/183192: What is the equation of the perpendicular bisector of the line between the points (2,2) and (6,6)? 1 solutions Answer 137550 by jim_thompson5910(28476)   on 2009-02-22 15:42:47 (Show Source): You can put this solution on YOUR website! Step 1) First find midpoint of the points (2,2) and (6,6) To find the midpoint, first we need to find the individual coordinates of the midpoint. X-Coordinate of the Midpoint: To find the x-coordinate of the midpoint, simply average the two x-coordinates of the given points by adding them up and dividing that result by 2 like this: So the x-coordinate of the midpoint is ----------------------------------------------------- Y-Coordinate of the Midpoint: To find the y-coordinate of the midpoint, simply average the two y-coordinates of the given points by adding them up and dividing that result by 2 like this: So the y-coordinate of the midpoint is So the midpoint between the points and is ======================================================= Step 2) Find the slope of the line through the points (2,2) and (6,6) Note: is the first point and is the second point . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is ========================================================== Step 3) Find the perpendicular slope Take the slope and flip the fraction (think of it as ) to get and change the sign to get . So the perpendicular slope is ========================================================== Step 4) Find the equation of the line with the perpendicular slope (found in step 3) which goes through the midpoint (found in step 1) To recap, the perpendicular slope is and the point that the perpendicular bisector goes through is (4,4) So let's find the equation of the line with a slope and goes through the point (4,4) If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add 4 to both sides to isolate y Combine like terms and to get So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is ============================================================== Answer: So the equation of the perpendicular bisector of the line between the points (2,2) and (6,6) is So the answer you're looking for is Here's the graph to verify the answer:

Rational-functions/183191: Find the coordinates of the point C, halfway between the points A(5,1) and B(-2, 7). 1 solutions Answer 137548 by jim_thompson5910(28476)   on 2009-02-22 15:31:26 (Show Source): You can put this solution on YOUR website! To find the midpoint, first we need to find the individual coordinates of the midpoint. X-Coordinate of the Midpoint: To find the x-coordinate of the midpoint, simply average the two x-coordinates of the given points by adding them up and dividing that result by 2 like this: So the x-coordinate of the midpoint is ----------------------------------------------------- Y-Coordinate of the Midpoint: To find the y-coordinate of the midpoint, simply average the two y-coordinates of the given points by adding them up and dividing that result by 2 like this: So the y-coordinate of the midpoint is So the midpoint between the points and is

Linear-equations/183190: Solve 5x + 6y=18 for y 1 solutions Answer 137545 by jim_thompson5910(28476)   on 2009-02-22 15:21:06 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from both sides. Rearrange the terms. Divide both sides by to isolate y. Break up the fraction. Reduce.

Expressions-with-variables/183181: expand. (x+y)^6 1 solutions Answer 137536 by jim_thompson5910(28476)   on 2009-02-22 14:35:18 (Show Source): You can put this solution on YOUR website! Start with the given expression To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle: 1    1   1    1   2   1    1   3   3   1    1   4   6   4   1    1   5   10   10   5   1    1   6   15   20   15   6   1    Looking at the row that starts with 1,6, etc, we can see that this row has the numbers: 1, 6, 15, 20, 15, 6, and 1 These numbers will be the coefficients of our expansion. So to expand , simply follow this procedure: Write the first coefficient. Multiply that coefficient with the first binomial term and then the second binomial term . Repeat this until all of the coefficients have been written. Once that has been done, add up the terms like this: Notice how the coefficients are in front of each term. However, we're not done yet. Looking at the first term , raise to the 6th power and raise to the 0th power. Looking at the second term raise to the 5th power and raise to the 1st power. Continue this until you reach the final term. Notice how the exponents of are stepping down and the exponents of are stepping up. So the fully expanded expression should now look like this: Distribute the exponents Multiply Multiply the terms with their coefficients So expands and simplifies to . In other words,

Polynomials-and-rational-expressions/183160: This question is from textbook 6g^3-24g^2+24g 1 solutions Answer 137533 by jim_thompson5910(28476)   on 2009-02-22 13:59:17 (Show Source): You can put this solution on YOUR website! I assume that you want to factor right? Please post full instructions. Start with the given expression Factor out the GCF Now let's focus on the inner expression ------------------------------------------------------------ Looking at we can see that the first term is and the last term is where the coefficients are 1 and 4 respectively. Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient -4? Let's list all of the factors of 4: Factors of 4: 1,2 -1,-2 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to 4 1*4 2*2 (-1)*(-4) (-2)*(-2) note: remember two negative numbers multiplied together make a positive number Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4 First Number Second Number Sum 1 4 1+4=5 2 2 2+2=4 -1 -4 -1+(-4)=-5 -2 -2 -2+(-2)=-4 From this list we can see that -2 and -2 add up to -4 and multiply to 4 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) note: is equivalent to since the term occurs twice. So also factors to ------------------------------------------------------------ So our expression goes from and factors further to ------------------ Answer: So factors to

Distributive-associative-commutative-properties/183091: 81x^4-16 1 solutions Answer 137483 by jim_thompson5910(28476)   on 2009-02-21 23:42:41 (Show Source): You can put this solution on YOUR website! Start with the given expression. Rewrite as . Rewrite as . Notice how we have a difference of squares where in this case and . So let's use the difference of squares formula to factor the expression: Start with the difference of squares formula. Plug in and . So this shows us that factors to . In other words . Now let's factor further ------------------------------------- Start with the given expression. Rewrite as . Rewrite as . Notice how we have a difference of squares where in this case and . So let's use the difference of squares formula to factor the expression: Start with the difference of squares formula. Plug in and . So this shows us that factors to . In other words . ---------------------------------------------------------------- Answer: So completely factors to

Distributive-associative-commutative-properties/183092: 36x²+12xy+y² 1 solutions Answer 137482 by jim_thompson5910(28476)   on 2009-02-21 23:40:03 (Show Source): You can put this solution on YOUR website! I assume that you want to factor. Looking at we can see that the first term is and the last term is where the coefficients are 36 and 1 respectively. Now multiply the first coefficient 36 and the last coefficient 1 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient 12? Let's list all of the factors of 36: Factors of 36: 1,2,3,4,6,9,12,18 -1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to 36 1*36 2*18 3*12 4*9 6*6 (-1)*(-36) (-2)*(-18) (-3)*(-12) (-4)*(-9) (-6)*(-6) note: remember two negative numbers multiplied together make a positive number Now which of these pairs add to 12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 12 First Number Second Number Sum 1 36 1+36=37 2 18 2+18=20 3 12 3+12=15 4 9 4+9=13 6 6 6+6=12 -1 -36 -1+(-36)=-37 -2 -18 -2+(-18)=-20 -3 -12 -3+(-12)=-15 -4 -9 -4+(-9)=-13 -6 -6 -6+(-6)=-12 From this list we can see that 6 and 6 add up to 12 and multiply to 36 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) note: is equivalent to since the term occurs twice. So also factors to ------------------------------------------------------------ Answer: So factors to

Quadratic_Equations/183093: 2x²-x=15 1 solutions Answer 137481 by jim_thompson5910(28476)   on 2009-02-21 23:37:41 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract 15 from both sides. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the answers are or