# -In the system of equations in x and y, 2x+3y=12, 4x+ay=16, where a is an integer, what would a have to be for the equations to be inconsistent? -In the system of equations in x and y, 2x+3y=12, 4x+ay=b, where a is the answer to question 2, what would b be if the equations are dependent?

-In the system of equations in x and y, 2x+3y=12, 4x+ay=16, where a is an integer, what would a have to be for the equations to be inconsistent?
-In the system of equations in x and y, 2x+3y=12, 4x+ay=b, where a is the answer to question 2, what would b be if the equations are dependent?
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Willie
Step 1
To find value of a such that the system of equations 2x+3y=124x+ay=16 is inconsistent.
Using this value of a, to find value of b such that the system of equations 2x+3y=124x+ay=b is dependent.
Step 2
Let the system of equations be
${a}_{1}x+{b}_{1}y+{c}_{1}=0$
${a}_{2}x+{b}_{2}y+{c}_{2}=0$
$\frac{{a}_{1}}{{a}_{2}}=\frac{{b}_{1}}{{b}_{2}}\ne \frac{{c}_{1}}{{c}_{2}}$
Thus, for given system of equations
$\frac{2}{4}=\frac{3}{a}\ne -\frac{12}{-16}$
$\frac{1}{2}=\frac{3}{a}$
a=6
Step 3
System of equations is dependent, if
$\frac{{a}_{1}}{{a}_{2}}=\frac{{b}_{1}}{{b}_{2}}=\frac{{c}_{1}}{{c}_{2}}$
Thus, for given system of equations
$\frac{2}{4}=\frac{3}{a}=-\frac{12}{-b}$
$=\frac{3}{6}=\frac{12}{b}$
$\frac{1}{2}=\frac{12}{b}$
b=24