Find the set of solutions for the linear system.
1 answer:
Answer:
The system has infinitely many solutions.
Step-by-step explanation:
To find the solution for this system of linear equations you must:
Step 1: Transform the augmented matrix to the reduced row echelon form.
A matrix is a rectangular arrangement of numbers into rows and columns.
A system of equations can be represented by an augmented matrix.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
This is matrix that represents the system
The augmented matrix can be transformed by a sequence of elementary row operations to the matrix.
There are three kinds of elementary matrix operations.
- Interchange two rows (or columns).
- Multiply each element in a row (or column) by a non-zero number.
- Multiply a row (or column) by a non-zero number and add the result to another row (or column).
Using elementary matrix operations, we get that
Row Operation 1: Multiply the 1st row by -1/3
Row Operation 2: Multiply the 2nd row by -1/3
Row Operation 3: Add 4/3 times the 2nd row to the 1st row
Step 2: Interpret the reduced row echelon form
The reduced row echelon form of the augmented matrix is
which corresponds to the system
We see that the variables can take arbitrary numbers; they are called free variables. Let
,
. All solutions of the system are given by
The system has infinitely many solutions.
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Answer:
Step-by-step explanation:
To sketch a quadratic function we need two things:
1) Nature of the curve
2) Vertex
3) y-intercept
The completing square form of the quadratic equation is:
where,
a represents the nature of the graph it can be maximum or minimum , meaning, if a > 0 then minimum(u shaped curve/happy face) and if a < 0 then maximum(n shaped curve/sad face).
h represents the x-coordinate of the vertex.
k represents the y-coordinate of the vertex.
Now if we compare g(x) with our completing square form we get the following:
When we simply compare the following we get ,
a = 1 , which means a > 0 since 1 is greater than 0 the nature of the curve will be minimum(happy face/u shaped)
h = 4, which means the x-coordinate of the vertex is 4
k = 12, which means the y-coordinate of the vertex is 12
Now we have the nature of the curve, we have the vertex now all we need is the y-intercept.
For y-intercept:
For y-intercept meaning at which point will the graph cross the y-axis(0 , y)
For that we expand the formula and turn it into the standard quadratic equation form by using the formula (a - b)^2
now we compare with the standard quadratic form:
here c is the y-intercept and while comparing we can see that c = 28 ,
so the curve cuts the y-axis at (0 , 28)
So we have all the three things that we need to graph our function.
So we just plot the y-intercept , the vertex , and join the dots. Just a tip draw a dotted line on the x-coordinate of the vertex because the vertex point is also called as a turning point where the graph goes in the opposite direction just like a mirror reflection. I attached 2 images you can check them out. One is handmade(i know i suck at drawing but still xD) , one is sketched by online graphing calculator.
