# -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?

Question
Equations
-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?

2021-01-20
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
$$\displaystyle{a}_{{1}}{x}+{b}_{{1}}{y}+{c}_{{1}}={0}$$
$$\displaystyle{a}_{{2}}{x}+{b}_{{2}}{y}+{c}_{{2}}={0}$$
$$\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}\ne\frac{{c}_{{1}}}{{c}_{{2}}}$$
Thus, for given system of equations
$$\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}\ne-\frac{{12}}{{-{{16}}}}$$
$$\displaystyle\frac{{1}}{{2}}=\frac{{3}}{{a}}$$
a=6
Step 3
System of equations is dependent, if
$$\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}=\frac{{c}_{{1}}}{{c}_{{2}}}$$
Thus, for given system of equations
$$\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}=-\frac{{12}}{{-{{b}}}}$$
$$\displaystyle=\frac{{3}}{{6}}=\frac{{12}}{{b}}$$
$$\displaystyle\frac{{1}}{{2}}=\frac{{12}}{{b}}$$
b=24

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