For the following parts, answer whether the given tuple describes a solution to the system of linear equations. If it is not a solution, explain why!
For the system,
\[\begin{aligned} 2x + y + 2z &= 3 \\ 3x + y - z &= 2 \end{aligned}\]is \((1, -1, 1)\) a solution?
No. Plugging the point into the second equation yields \(3+1-1=2\) which is false, since \(3 \ne 2\).
For the system,
\[\begin{aligned} 5x - y + z &= 7 \\ 3x + y - z &= 1 \end{aligned}\]is \((1, s-2, s)\) a solution (for every real number \(s\))?
Yes. Plugging the point into the first equation yields \(7=7\) and the second equation yields \(1=1\), both of which are true.
Rewrite the following system of equations as an augmented matrix. You do not actually have to solve the system.
\[\begin{aligned} 5x - y + z + 6w &= 7 \\ 3x + 3y - 2z &= 1 \\ y - z &= 0 \\ x - y - z &= 0 \\ y &= 3 \end{aligned}\]\[\begin{pmatrix} 5 & -1 & 1 & 6 & 7 \\ 3 & 3 & -2 & 0 & 1 \\ 0 & 1 & -1 & 0 & 0 \\ 1 & -1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 3 \end{pmatrix}\]
Draw a picture that describes a system of three linear equations that has no solution.
There are many solutions. Here is one.
Find all solutions to the following system of linear equations:
\[\begin{aligned} 3x + y &= 1 \\ x - y &= 2 \end{aligned}\]As an augmented matrix, the system reduces to:
\[\begin{pmatrix} 1 & 0 & 3/4 \\ 0 & 1 & -5/4 \\ \end{pmatrix}\]So, \(x = 3/4\) and \(y = -5/4\).
For the following parts, answer whether the function described is linear or not.
\(f(x) = 3x + 1\)
\(f(x_1, x_2) = 3x_1 - 6x_2\)
\(f(x_1, x_2, x_3) = 3x_1 + x_3\)
\(f(x, y) = 3xy + x + y\)
Originally, we had to solve this using the form of a linear function. Now, we can use the definition.
No, as \(f(2) = 7 \ne 2f(1) = 2(4)=8\).
Yes, as \(f(a+b,c+d) = 3(a+b)-6(c+d) = (3a-6c)+(3b-6d) = f(a,c)+f(b,d)\).
Yes, as \(f(a+b,c+d,x+y) = 3(a+b)+(x+y) = (3a+x) + (3b+y) = f(a,c,x) + f(b,d,y)\).
No, as \(f(2,2) = 16 \ne 2f(1,1) = 2(5) = 10\).