Algebra

# System of Linear Equations - 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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