Chapter Closure
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Exercise 66 Page 40
To find the missing numbers, let's identify the variables using our variable diamond and substitute the correct numbers into our pattern.
In this case we are given the values of the product xy and the sum x+y. We want to find the values of x and y.
Notice that the pair x=4 and y=2 also works for the pattern. In this situation we have two possibilities to complete the diamond.
Here, we are given the values for x and y and we want to complete the diamond.
x=-2 y=6To do this we can substitute the numbers into the above pattern.
xy & (I) x+y & (II)
(I), (II): x= -2, y= 6
-2 * 6 -2+ 6
-12 2 4
Now, we can complete our diamond.
Here we are given that x=4 and xy=-28. Substituting these values into our pattern gives us the following equations.
4y&=-28
4+y&= ?
From the first equation we can calculate y by isolating it on one side.
4y=-28
▼
.LHS /4.=.RHS /4.
DivEqn
.LHS /4.=.RHS /4.
4y/4=-28/4
MoveNegNumToFrac
Put minus sign in front of fraction
4y/4=- 28/4
SplitIntoFactors
Split into factors
4* y/4=- 7* 4/4
CancelCommonFac
Cancel out common factors
y=-7
We found the value of y. Now, we substitute it into the second equation to find the square at the bottom.
Finally, we can complete our diamond.
Here, we are given that y=-1 and x+y=-4. Substituting these values into our pattern gives us the following equations.
x(-1) &= ?
x+(-1)&= -4
From the second equation we can calculate x by isolating it on one side.
We found the value of x. Now, we substitute it into the first equation to find the square at the top.
Now we can complete our diamond.
Here, we are given the values for x and y and we want to complete the diamond.
x=3.25 y=4To do this we can substitute the numbers into the above pattern.
xy & (I) x+y & (II)
(I), (II): x= 3.25, y= 4
3.25 * 4 3.25+ 4
13 7.25
Now, we can complete our first diamond.
In this case we are given the values of the product xy and the sum x+y. We want to find the values of x and y.
xy&=24
x+y&=-11
To find these values, let's think of different pairs of numbers which multiply to give us 24. Since the product of x and y is positive, we need a pair with the same sign — two positive numbers or two negative numbers. Let's also check their sums to know which, if any of these, will match the second condition as well.
c|ccc
&Product&Sum
1 and 24 &24&25& *
2 and 12 &24&14& *
3 and 8 &24&11& *
4 and 6 &24&10& *
- 1 and - 24 &24&- 25& *
- 2 and - 12 &24&- 14& *
- 3 and - 8 &24&-11& ✓
- 4 and - 6 &24&- 10& *
As we can see, the pair x=-3 and y=-8 works for the pattern. Therefore, this is the correct pair of numbers to complete the diamond.
Notice that the pair x=-8 and y=-3 also works for the pattern. In this situation we have two possibilities to complete the diamond.
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