Why Are Coefficients of x Plotted on the y-Axis in Linear Combinations?

[mirza21]

I have to linear equations

3x + 2y =7 and -6x + 6y= 6

when expressed as linear combination in column vector form they become:

x[3,-6] + y[2,6] = [7,6]

when solving this linear combination graphically ,First vector is plotted
like this x=3, y=-6 and second vector is plotted like x=2, y=6 as per
textbook

1) first vector contains x coeffceints of equation 1 and 2. I am consfused why second element of first vector(-6) is plotted on y axis. It is x element as per equation 2. Please clarify on this why this done like
this?

 

[Fredrik]

I assume the problem is to solve the equation that you write as x[3,-6] + y[2,6] = [7,6]. Suppose you had been asked to solve the equation u[3,-6] + v[2,6] = [7,6] instead. Would that have been a different problem?

 

[HallsofIvy]

In other words, the "x" and "y" used as coefficients for the vectors are NOT the "x" and "y" labeling the axes. It would, indeed, be better not to use "x" and "y" in the equation as Fredrick suggests.

 

[Noxide]

Each component of a vector can be though of as a "new axis", much like the x and y axes.
Take a vector (a, b, c, d)
In the coordinate system you are familiar with, a would be the x-axis value, b the y-axis, c the z-axis, d the t-axis... etc and you can go to as many axes as you want

so even though you see an x in front of (3, -6) it does not mean that every component in that vector is on the x-axis it just means that x is some quantity multiplied by each component that lies on it's own independent axis

 

[blue_raver22]

I understand your confusion about the linear combination of two equations. Let me explain it to you in a more mathematical way.

A linear combination is a mathematical operation where two or more equations are added, subtracted, or multiplied by a constant to form a new equation. In this case, we have two equations:

3x + 2y = 7 and -6x + 6y = 6

When we express these equations in column vector form, we get:

[3, 2] and [-6, 6]

The first element in each vector represents the coefficient of x, while the second element represents the coefficient of y. So, for the first equation, the coefficient of x is 3 and the coefficient of y is 2. Similarly, for the second equation, the coefficient of x is -6 and the coefficient of y is 6.

Now, when we solve the linear combination graphically, we plot these two vectors on a graph. The first vector is plotted with the x-coordinate as 3 and the y-coordinate as 2. This is represented as (3,2) on the graph. Similarly, the second vector is plotted with the x-coordinate as -6 and the y-coordinate as 6. This is represented as (-6,6) on the graph.

The reason why we plot the second element of the first vector (-6) on the y-axis is because in the column vector form, the second element represents the coefficient of y. So, we plot it on the y-axis to show its relation to the y-variable. This is just a visual representation to help us understand the linear combination better.

I hope this clarifies your confusion about the linear combination. Remember, in a linear combination, the coefficients of x and y are represented by the first and second elements in the column vectors, respectively. Keep practicing and you will become more comfortable with this concept. Good luck!