Algebra

System of Linear Equations

System of Linear Equations - Problem Solving

         

Find the sum of all values of constant kk such that there exists non-zero solutions (i.e x0x \neq 0 and y0y \neq 0) to the following equations: 4x+y=kx,      12x+15y=ky.4x+y=kx, \;\;\; 12x+15y=ky.

Let Max(x,y)\text{Max}(x, y) and Min(x,y)\text{Min}(x, y) be defined as follows for real numbers xx and y:y: Max(x,y)={x(xy)y(x<y),Min(x,y)={x(x<y)y(xy). \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 xx and y,y, what is 3xy:3xy: Max(x,y)=5x2y+79,Min(x,y)=4x+3y47?\text{Max}(x, y)=5x-2y+79, \quad \text{Min}(x, y)=4x+3y-47?

Consider the system of linear equations 3x+Ay=108 andx2+y3=B,\begin{aligned} 3x+Ay &= -108 \ \mbox{and} \\ -\frac{x}{2}+\frac{y}{3} &= B, \end{aligned} where AA and BB are constants. If this system has an infinite number of solutions, what is the value of BAB-A?

Let x=ax=a and y=by=b be the solution of the simultaneous equations 5x312+7y+68=1\frac{5x-3}{12}+\frac{7y+6}{8}=1 and (x+53)D=2y6 \frac{(x+53)}{D} = \frac{2y}{6} If b=a+5b=a+5, what is the value of the constant DD?

If x=Ax=A and y=By=B are the solutions of the simultaneous equations 3y4x+162=3xy3=2xy,\frac{3y-4x+16}{2}=\frac{3x-y}{3}=2x-y, what is the value of A+BA+B?

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