Question

Write a linear equation for the tables shown. y = mx + b y =__ x +____

x y x y
-6 12 -14 3
-5 14 -12 6
-4 16 -10 9
-3 18 - 8 12
-2 20 - 6 15

Answer 1

Answers:

1) First table: y=2x+24

2) Second table: y=(3/2)x+24

Solution:

m=(y2-y1)/(x2-x1)

y-y1=m(x-x1)

1) First table. Taking the first two points of the table:

P1=(-6,12)=(x1,y1)→x1=-6, y1=12

P2=(-5,14)=(x2,y2)→x2=-5, y2=14

m=(y2-y1)/(x2-x1)

m=(14-12)/(-5-(-6))

m=(2)/(-5+6)

m=(2)/(1)

m=2

y-y1=m(x-x1)

y-12=2(x-(-6))

y-12=2(x+6)

y-12=2x+2(6)

y-12=2x+12

y-12+12=2x+12+12

y=2x+24

2) Second table. Taking the first two points of the table:

P1=(-14,3)=(x1,y1)→x1=-14, y1=3

P2=(-12,6)=(x2,y2)→x2=-12, y2=6

m=(y2-y1)/(x2-x1)

m=(6-3)/(-12-(-14))

m=(3)/(-12+14)

m=(3)/(2)

m=3/2

y-y1=m(x-x1)

y-3=(3/2)(x-(-14))

y-3=(3/2)(x+14)

y-3=(3/2)x+(3/2)(14)

y-3=(3/2)x+(3)(14)/2

y-3=(3/2)x+42/2

y-3=(3/2)x+21

y-3+3=(3/2)x+21+3

y=(3/2)x+24

Answer 2

ANSWER TO QUESTION 1

`y=2x+24`

Step-by-step explanation

Method 1: Finding the equation given any 2 points

We choose any two of the ordered pairs from the first table, say

`(-2,20)`

and

`(-3,18)`

We determine the slope using the formula;

`Slope=(y_2-y_1)/(x_2-x_1)`

Let
`(x_1,y_1)` be
`(-2,20)`

and

`(x_2,y_2)` be
`(-3,18)`

Then,

`Slope=(18-20)/(-3--2)`

`\Rightarrow Slope=(18-20)/(-3+2)`

`\Rightarrow Slope=(-2)/(-1)`

`\Rightarrow Slope=2`

We now use the formula,

`y-y_1=m(x-x_1)` to find the equation of this line.

That is ;

`y-20=2(x--2)`

`y-20=2(x+2)`

`y=2x+4+20`

`y=2x+24`

ANSWER TO QUESTION 2

Method 2: Using Simultaneous equations

We use the slope intercept form for the second table

`y=mx+b`

The point

`(-6,15)` must satisfy this line.

`15=-6m+b--(1)`

The point

`(-8,12)` must also satisfy this line.

`12=-8m+b--(2)`

Equation (1) minus Equation (2) gives

`2m=3`

`\Rightarrow m=(3)/(2)`

We substitute
`m=(3)/(2)` in to equation (1)

and solve for b.

`\Rightarrow 15=-6* (3)/(2)+b`

`\Rightarrow 15=-9+b`

`\Rightarrow 15+9=b`

`\Rightarrow 24=b`

Solving simultaneously gives

`b=24` and
`m=(3)/(2)`

Hence the equation is

`y=(3)/(2)x+24`