Exercise 83 Page 80

a Comparing this diamond to the one showing the pattern, we can create two equations.

y = x + y = - 1 10

Since we know that

y = - 1,

we can substitute it in the second equation to find

x .

x + y = 10

Substitute

$y = - 1$

x + (- 1) = 10

AddNeg

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

x - 1 = 10

AddEqn

$LHS + 1 = RHS + 1$

x = 11

Since we know that

x = 11,

we can find the upper square of the diamond by substituting

x = 11

and

y = - 1

in the expression

x y .

x y

SubstituteII

$x = 11$ , $y = - 1$

11 (- 1)

MultPosNeg

$a (- b) = - a \cdot b$

- 11

Now we can complete our diamond!

b In this diamond we have been given the values of

x

and

y .

x = 5 y = - 3

Let's determine

x y

and

x + y

by substituting these values into the expressions.

x y : x + y : 5 (- 3) = - 15 5 + (- 3) = 2

Now we can complete our diamond!

c With the given information we can create the following equations.

x = x y = \frac{2}{3} 1

To complete the diamond we should substitute

x = \frac{2}{3}

in the second equation and find

y .

x y = 1

Substitute

$x = \frac{2}{3}$

\frac{2}{3} \cdot y = 1

▼

Simplify

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{2 y}{3} = 1

MultEqn

$LHS \cdot 3 = RHS \cdot 3$

2 y = 3

DivEqn

$LHS / 2 = RHS / 2$

y = \frac{3}{2}

Having found

y,

we can determine the last square in the diamond by substituting

x

and

y

x + y .

x + y

SubstituteII

$x = \frac{2}{3}$ , $y = \frac{3}{2}$

\frac{2}{3} + \frac{3}{2}

▼

Simplify

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

\frac{4}{6} + \frac{3}{2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

\frac{4}{6} + \frac{9}{6}

AddFrac

Add fractions

\frac{1 3}{6}

Finally, we can complete our diamond!

d For the last diamond we can create the following equations.

x y = x + y = - 14 - 5

This tells us that

x

and

y,

when multiplied, must give us a negative number. Therefore, either

x

y

must be negative and the other one must be positive. We have two cases that both results in a product of

- 14

when multiplied.

- 14 = - 2 (7) - 14 = 2 (- 7)

To determine which pair of integers is correct, we have to check which one sums to

- 5 .

- 2 + 7 2 + (- 7) = 5 = - 5 \times ✓

As we can see, the values of

x

and

y

are

2

and

- 7 .

Now we can complete our diamond!

Subchapter links

2.3.1 Finding an Equation Given the Slope and a Point p.80

2.3.2 Finding the Equation of a Line Through Two Points p.82-83

Extension Activity Finding y=mx+b from Graphs and Tables p.85

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