a 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.
xy&=- 80
x+y&=2
To find these values, let's think of different pairs of numbers which multiply to give us - 80.
Since the product of x and y is negative, we need a pair with one negative and one positive number.
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 - 80 &- 80&- 79& *
2 and - 40 &- 80&- 38& *
4 and - 20 &- 80&- 16& *
5 and - 16 &- 80&- 11& *
8 and - 10 &- 80&- 2& *
- 1 and 80 &- 80&79& *
- 2 and 40 &- 80&38& *
- 4 and 20 &- 80&16& *
- 5 and 16 &- 80&11& *
- 8 and 10 &- 80&2& ✓
As we can see, the pair x=- 8 and y=10 works for the pattern. Therefore, this is the correct pair of numbers to complete the diamond.
Notice that the pair x=10 and y=- 8 also works for the pattern. In this situation we have two possibilities to complete the diamond.
b Here, just like in Part A, 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&=12
x+y&=- 7
To find these values, let's think of different pairs of numbers which multiply to give us 12.
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 12 &12&13& *
2 and 6 &12&8& *
3 and 4 &12&7& *
- 12 and - 1 &12&- 13& *
- 6 and - 2 &12&- 8& *
- 4 and - 3 &12&- 7& ✓
As we can see, the pair x=- 3 and y=- 4 works for the pattern. Therefore, this is the correct pair of numbers to complete the diamond.
Notice that the pair x=- 4 and y=- 3 also works for the pattern. In this situation we have two possibilities to complete the diamond.
c Here 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&=0
x+y&=7
Now, let's think of different pairs of numbers which multiply to give us 0. Since any number multiplied by 0 is always 0, and a product of any non-zero numbers is a non-zero number, we know that one of the numbers — let's say x — equals 0.
Using this information we can simplify our system of equations.
(0)y&=0
0+y&= 7
Notice that from the second equation we can calculate y by isolating it on one side.
As we can see, the pair x=0 and y=7 works for the pattern. Therefore, this is the correct pair of numbers to complete the diamond.
Notice that the pair x=7 and y=0 also works for the pattern. In this situation we have two possibilities to complete the diamond.
d Here 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&=- 81
x+y&=0
Notice that since the sum of x and y is 0, it means that they are opposite numbers, y=- x. We can substitute it into our system of equations.
x(- x)&=- 81
x+(- x)&=0
Let's find x from the first equation by isolating it on one side.
We found that x=9 or x=- 9. Therefore, we have two possibilities to complete the diamond. One of them contains the pair x=9 and y= - x = - 9.
The second diamond, which also works for the pattern, contains the pair x=- 9 and y=- x = 9.
e This time, to avoid confusion, we will call the numbers we are looking for a and b, instead of x and y. Here, we are given the values of the product ab and the sum a+b. We want to find the values of a and b.
ab&=6x^2
a+b&=5x
To find these values, let's think of different pairs of expressions whose product is 6x^2.
Since the product and the sum of a and b have positive coefficients, we need a pair of two expressions with positive coefficients.
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 6x^2 &6x^2&6x^2+1& *
2 and 3x^2&6x^2&3x^2+2& *
3 and 2x^2 &6x^2&2x^2+3& *
6 and x^2 &6x^2&x^2+6& *
x and 6x &6x^2&7x& *
2x and 3x &6x^2&5x& ✓
As we can see, the pair a=2x and b=3x works for the pattern. Therefore, this is the correct pair of expressions to complete the diamond.
Notice that the pair a=3x and b=2x also works for the pattern. In this situation we have two possibilities to complete the diamond.
f This time, to avoid confusion we will call the numbers we are looking for a and b, instead of x and y. Here, we are given the values of the product ab and the sum a+b. We want to find the values of a and b.
ab&=- 7x^2
a+b&=- 6x
Let's now think of different pairs of expressions which multiply to give us - 7x^2.
Since the product and the sum of a and b have negative coefficients, we need a pair of expressions with one positive and one negative coefficient.
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 - 7x^2 &- 7x^2&- 7x^2+1& *
- 1 and 7x^2 &- 7x^2&7x^2-1& *
x and - 7x &- 7x^2&- 6x& ✓
- x and 7x &- 7x^2&6x& *
As we can see, the pair a=x and b=- 7x works for the pattern. Therefore, this is the correct pair of numbers to complete the diamond.
Notice that the pair a=- 7x and b=x also works for the pattern. In this situation we have two possibilities to complete the diamond.
