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3. Compare the ratios [tex]\frac{a_1}{a_2}[/tex], [tex]\frac{b_1}{b_2}[/tex], and [tex]\frac{c_1}{c_2}[/tex] to determine whether the following pair of linear equations is consistent.



Answer :

GinnyAnswer
Certainly! Let's analyze the given problem step-by-step.

We need to compare the ratios [tex]\(\frac{a_1}{a_2}\)[/tex], [tex]\(\frac{b_1}{b_2}\)[/tex], and [tex]\(\frac{c_1}{c_2}\)[/tex].

Given values are:
- [tex]\(a_1 = 1\)[/tex], [tex]\(a_2 = 2\)[/tex]
- [tex]\(b_1 = 2\)[/tex], [tex]\(b_2 = 3\)[/tex]
- [tex]\(c_1 = 3\)[/tex], [tex]\(c_2 = 4\)[/tex]

First, we calculate the ratio [tex]\(\frac{a_1}{a_2}\)[/tex]:
[tex]\[ \frac{a_1}{a_2} = \frac{1}{2} = 0.5 \][/tex]

Next, we calculate the ratio [tex]\(\frac{b_1}{b_2}\)[/tex]:
[tex]\[ \frac{b_1}{b_2} = \frac{2}{3} \approx 0.6666666666666666 \][/tex]

Then, we calculate the ratio [tex]\(\frac{c_1}{c_2}\)[/tex]:
[tex]\[ \frac{c_1}{c_2} = \frac{3}{4} = 0.75 \][/tex]

So, the ratios are:
[tex]\[ \frac{a_1}{a_2} = 0.5 \][/tex]
[tex]\[ \frac{b_1}{b_2} \approx 0.6666666666666666 \][/tex]
[tex]\[ \frac{c_1}{c_2} = 0.75 \][/tex]

By comparing these ratios, we can conclude the numerical relationships among them:
- [tex]\(\frac{a_1}{a_2}\)[/tex] (0.5) is less than [tex]\(\frac{b_1}{b_2}\)[/tex] (approximately 0.6666666666666666).
- [tex]\(\frac{b_1}{b_2}\)[/tex] (approximately 0.6666666666666666) is less than [tex]\(\frac{c_1}{c_2}\)[/tex] (0.75).

Therefore, we have the order:
[tex]\[ \frac{a_1}{a_2} < \frac{b_1}{b_2} < \frac{c_1}{c_2} \][/tex]

This is the result of comparing the given ratios.

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