Question

choose all tables where the ratios of ordered pairs represent a proportional relationship between x and y.
a.

x	y
4	7
8	14
12	21

b.

x	y
3	4
6	12
9	16

c.

x	y
6	8
18	16
24	24

d.

x	y
1	7
2	14
3	21

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship between \(x\) and \(y\) means \(y = kx\) (where \(k\) is constant), so \(\frac{y}{x}\) should be constant for all non - zero \(x\), and when \(x = 0\), \(y = 0\).

Step2: Analyze Table A

• For \(x = 4,y = 7\), \(\frac{y}{x}=\frac{7}{4}=1.75\)
• For \(x = 8,y = 14\), \(\frac{y}{x}=\frac{14}{8}=1.75\)
• For \(x = 12,y = 21\), \(\frac{y}{x}=\frac{21}{12}=1.75\)
• When \(x = 0,y = 0\). So the ratio is constant.

Step3: Analyze Table B

• For \(x = 3,y = 4\), \(\frac{y}{x}=\frac{4}{3}\approx1.33\)
• For \(x = 6,y = 12\), \(\frac{y}{x}=\frac{12}{6}=2\)
• The ratios are not equal, so it is not proportional.

Step4: Analyze Table C

• For \(x = 6,y = 8\), \(\frac{y}{x}=\frac{8}{6}=\frac{4}{3}\approx1.33\)
• For \(x = 18,y = 16\), \(\frac{y}{x}=\frac{16}{18}=\frac{8}{9}\approx0.89\)
• The ratios are not equal, so it is not proportional.

Step5: Analyze Table D

• For \(x = 1,y = 7\), \(\frac{y}{x}=7\)
• For \(x = 2,y = 14\), \(\frac{y}{x}=\frac{14}{2}=7\)
• For \(x = 3,y = 21\), \(\frac{y}{x}=\frac{21}{3}=7\)
• When \(x = 0,y = 0\). So the ratio is constant.

Answer:

A, D