Question

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solve a system of equations to find the two numbers described.
1.. The sum of two numbers is 17 and their difference is 29.
2. The sum of a number and twice a greater number is 8. The sum of the greater number and twice the lesser number is -6.

Answer 1

For statement 1:

y = 23; x = -6

For statement 2:

`y =(22)/(3)`;
`x = -(20)/(3)`

Explanation:

Let x = Lesser number

y = Greater number

According to statement 1:

x + y = 17

y - x = 29

The above two equations are simultaneous linear equations, which can be solved either by substitution, elimination or graphical method:

Elimination method:

The coefficients of 'x' or 'y' need to be the same with equal signs. The coefficients of 'x' will be focused on:

x + y = 17

-x + y = 29

In the net total, the 'x' terms will cancel each other :

y + y = 17 + 29

2y = 46

`y = (46)/(2)`

y = 23

Substituting this value of y in any of the equations to calculate the value of x:

x + 23 = 17

x = 17 - 23

x = -6

According to statement 2:

x + 2y = 8

y + 2x = -6

Rearranging the equations and focusing on the coefficients of 'x':

(x + 2y = 8) Multiplied by 2

(2x + y = -6) Multiplied by -1

2x + 4y = 16

-2x - y = 6

In the net total, the 'x' terms will cancel each other:

4y - y = 16 + 6

3y = 22

`y = (22)/(3)`

Substitute the above calculated value of 'y' in one of the equations to determine the value of 'x':

`2x + 4((22)/(3)) = 16`

`2x + (88)/(3) = 16`

`2x = 16 -(88)/(3)`

`2x = (48)/(3) - (88)/(3)`

`2x = -(40)/(3)`

`x = -(40)/((3)(2))`

`x = -(20)/(3)`