Exercise 145 Page 307

Practice makes perfect

We want to complete the given Diamond Problem. Let's consider the given diagram and the pattern used in the problem!

Notice that we are given

y

is equal to

6 .

We can fill the remaining corners when we find a number corresponding to an arbitrary variable

x

such that the product

x y

is equal to

0 .

x y = 0 \Rightarrow x (6) = 0

Recall that the Zero Product Property says that the product of two real numbers is

0

when one or both of the numbers equal

0 .

That must mean, in our case,

x

is equal to

0 .

Now we can calculate the value of the sum

(x + y) .

x + y

SubstituteII

$x = 6$ , $y = 0$

6 + 0

IdPropAdd

Identity Property of Addition

6

Finally, we can complete our Diamond Problem.

Let's consider the given Diamond Problem and the pattern used in the problem!

We can see on the diagram that $x$ is $8$ and $y$ is $- 2 .$ We can fill the remaining parts of the diamond by calculating the product and the sum of the factors.

	$x = 8,$ $y = - 2$
Product $(x y)$	$8 (- 2) = - 16$
Sum $(x + y)$	$8 + (- 2) = 6$

Now we can complete our Diamond Problem.

We want to complete the given Diamond Problem. Let's look at the diagram!

Notice that the arbitrary variable

y

has a value of

- 4 .

We also know that the sum

(x + y)

is equal to

0 .

Let's use this information to find the value of

x!

x + y = 0

Substitute

$y = - 4$

x + (- 4) = 0

AddEqn

$LHS + 4 = RHS + 4$

x + (- 4) + 4 = 0 + 4

▼

Simplify

CommutativePropAdd

Commutative Property of Addition

x + 4 + (- 4) = 0 + 4

AddNeg

$a + (- b) = a - b$

x + 4 - 4 = 0 + 4

DiffZero

$4 - 4 = 0$

x + 0 = 0 + 4

IdPropAdd

Identity Property of Addition

x = 4

The variable

x

is equal to

4 .

Next, we can calculate the value of the product,

x y .

x y

SubstituteII

$x = 4$ , $y = - 4$

4 (- 4)

Multiply

- 16

Now we can complete our Diamond Problem.

We want to complete the given Diamond Problem. Let's consider the given diagram and the pattern used in the problem!

We can fill the missing parts of the Diamond by finding two rational numbers that sum up to $15.2$ and have a product of $52 .$

Product $(x y)$	Sum $(x + y)$	Is Sum Equal to $15.2 ?$
$52 = 0.52 (100)$	$0.52 + 100 = 100.52$	$\times$
$52 = 5.2 (10)$	$5.2 + 10 = 15.2$	$✓$
$52 = 52 (1)$	$52 + 1 = 53$	$\times$

Notice that our Diamond Problem corresponds to the pattern when one of the variables is equal to $5.2$ and the other is equal to $10 .$ Let's fill our Diamond Problem!