Question

A number is equal to the sum of half a second number and 3. The first number is also equal to the sum of one-

quarter of the second number and 5. The situation can be represented by using the graph below, where x
represents the second number.
16
14
12+
10
2-
4-22-
2 4 6 8 10 12 14 16 X
Which equations represent the situation?
8
64

Answer 1

The equations are

If first no be x and second no be y

`\\ \rm\Rrightarrow x=(y)/(2)+3`

Simplify

`\\ \rm\Rrightarrow x=(y+6)/(2)`

`\\ \rm\Rrightarrow y+6=2x`

`\\ \rm\Rrightarrow 2x-y-6=0\dots (1)`

Second equation

`\\ \rm\Rrightarrow x=(1)/(4)y+5`

`\\ \rm\Rrightarrow x=(y+20)/(4)`

`\\ \rm\Rrightarrow y+20=4x`

`\\ \rm\Rrightarrow 4x-y-20=0\dots(2)`

Graphs attached

The numbers are 7 and 8

Answer 2

`\sf y=(1)/(2)x+3\quad and \quad y=(1)/(4)x+5`

Explanation:

The question clearly states "where x represents the second number".

Therefore:

• Let y = first number
• Let x = second number

Given:

• A number is equal to the sum of half a second number and 3

`\sf \implies y=(1)/(2)x+3`

Given:

• The first number is equal to the sum of one-quarter of the second number and 5

`\sf \implies y=(1)/(4)x+5`

Therefore, the equations that represent the situation are:

`\sf y=(1)/(2)x+3\quad and \quad y=(1)/(4)x+5`

Proof

Solve the system of equations. Equate the equations and solve for x:

`\implies \sf (1)/(2)x+3=(1)/(4)x+5`

`\implies \sf (1)/(4)x=2`

`\implies \sf x=8`

Substitute the found value of 8 into one of the equations and solve for y:

`\sf \implies y=(1)/(2)(8)+3`

`\sf \implies y=4+3`

`\sf \implies y=7`

Therefore, the solution to the system of equations is (8, 7).

The solution to the system of equations is shown on the graph as the point of intersection. The point of intersection is (8, 7), hence the found equations are correct.