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Systems of Linear Equations

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Problem Solving

         

Find the sum of all values of constant \(k\) such that there exists non-zero solutions (i.e \(x \neq 0\) and \(y \neq 0\)) to the following equations: \[4x+y=kx, \;\;\; 12x+15y=ky.\]

Let \(\text{Max}(x, y)\) and \(\text{Min}(x, y)\) be defined as follows for real numbers \(x\) and \(y:\) \[ \text{Max}(x, y)=\begin{cases} x \quad (x\ge y) \\ y \quad (x<y), \\ \end{cases}\\ \text{Min}(x, y)=\begin{cases} x \quad (x< y) \\ y \quad (x\ge y). \end{cases} \]

If the following holds for distinct numbers \(x\) and \(y,\) what is \(3xy:\) \[\text{Max}(x, y)=5x-2y+79, \quad \text{Min}(x, y)=4x+3y-47?\]

Consider the system of linear equations \[\begin{align} 3x+Ay &= -108 \ \mbox{and} \\ -\frac{x}{2}+\frac{y}{3} &= B, \end{align}\] where \(A\) and \(B\) are constants. If this system has an infinite number of solutions, what is the value of \(B-A\)?

Let \(x=a\) and \(y=b\) be the solution of the simultaneous equations \[\frac{5x-3}{12}+\frac{7y+6}{8}=1\] and \[ \frac{(x+53)}{D} = \frac{2y}{6}\] If \(b=a+5\), what is the value of the constant \(D\)?

If \(x=A\) and \(y=B\) are the solutions of the simultaneous equations \[\frac{3y-4x+16}{2}=\frac{3x-y}{3}=2x-y,\] what is the value of \(A+B\)?

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