Exercise 19 Page 354

Let's call the first number

a

and the second number

b .

The phrase the sum of two numbers is

22

can be written algebraically by equating

22

with the sum of

a

and

b .

a + b = 22

The second phrase, their difference is

12

, can be written algebraically by equating

12

with the difference

a - b .

a - b = 12

Combining these equations, we get a system of equations.

{a + b = 22 a - b = 12 (I) (II)

Since

b

and

- b

have opposite coefficients, we will solve this system using elimination by addition.

{a + b = 22 a - b = 12 (I) (II)

$(II):$ Add $(I)$

{a + b = 22 a - b + a + b = 12 + 22

$(II):$ Add terms

{a + b = 22 2 a = 34

$(II):$ $LHS / 2 = RHS / 2$

{a + b = 22 a = 17

Now that we have found the value of

a,

we can substitute it into Equation (I) to solve for

b .

{a + b = 22 a = 17

$(I):$ $a = 17$

{17 + b = 22 a = 17

$(I):$ $LHS - 17 = RHS - 17$

{b = 5 a = 17

We have found that the numbers are

5

and

17 .