Given the system of equations:
ax + by = -12
7x + 8y = 6
Let's determine the values of a and b that make the system dependent.
Since both equations are dependent, we have:
ax + by = -12
7x + 8y = 6
Apply the formula:`
[tex]\frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}[/tex]
Where:
a1 = a
b1 = b
c1 = -12
a2 = 7
b2 = 8
c2 = 6
Thus, we have:
[tex]\frac{a}{7}=\frac{b}{8}=\frac{-12}{6}[/tex]
Thus, for a, consider:
[tex]\begin{gathered} \frac{a}{7}=\frac{-12}{6} \\ \\ \text{Cross mutiply:} \\ 6a=-12(7) \\ \\ 6a=-84 \\ \\ \text{Divide both sides by 6:} \\ \frac{6a}{6}=\frac{-84}{6} \\ \\ a=-14 \end{gathered}[/tex]
For the value of b, we have:
[tex]\begin{gathered} \frac{b}{8}=\frac{-12}{6} \\ \\ \text{Cross multiply:} \\ 6b=-12(8) \\ \\ 6b=-96 \\ \\ \text{Divide both sides by 6:} \\ \frac{6b}{6}=\frac{-96}{6} \\ \\ b=-16 \end{gathered}[/tex]
Therefore, the values of a and b that make the system dependent are:
a = -14
b = -16
ANSWER:
a = -14, b = -16
