Question

1. determine an equation that models the relationship between each set of numbers.

a)

x	y
0	12.5
1	14.4
2	16.3
3	18.2

b)

a	b
1	2
2	0.5
3	-1.0
4	-2.5

c)

v	t
0	10.0
5	10.5
10	11.0
15	11.5

d)

m	p
1	-6.8
2	-1.2
3	4.4
4	10.0

1. competency

Explanation:

Step1: Check linearity (constant Δy/Δx)

For a): $\Delta y = 14.4-12.5=1.9$, $16.3-14.4=1.9$, $18.2-16.3=1.9$; $\Delta x=1$. Slope $m=1.9$. Y-intercept $b=12.5$ (when $x=0$).
Equation: $y=1.9x+12.5$

Step2: Check linearity (constant Δb/Δa)

For b): $\Delta b = 0.5-2=-1.5$, $-1.0-0.5=-1.5$, $-2.5-(-1.0)=-1.5$; $\Delta a=1$. Slope $m=-1.5$. Use $(a=1,b=2)$: $2=-1.5(1)+b \implies b=3.5$.
Equation: $b=-1.5a+3.5$

Step3: Check linearity (constant Δt/Δv)

For c): $\Delta t = 10.5-10.0=0.5$, $11.0-10.5=0.5$, $11.5-11.0=0.5$; $\Delta v=5$. Slope $m=\frac{0.5}{5}=0.1$. Y-intercept $b=10.0$ (when $v=0$).
Equation: $t=0.1v+10.0$

Step4: Check linearity (constant Δp/Δm)

For d): $\Delta p = -1.2-(-6.8)=5.6$, $4.4-(-1.2)=5.6$, $10.0-4.4=5.6$; $\Delta m=1$. Slope $m=5.6$. Use $(m=1,p=-6.8)$: $-6.8=5.6(1)+b \implies b=-12.4$.
Equation: $p=5.6m-12.4$

Answer:

a) $y=1.9x+12.5$
b) $b=-1.5a+3.5$
c) $t=0.1v+10.0$
d) $p=5.6m-12.4$