5.1: System of Linear Equations

In this course we will spend a lot of time working with systems of linear equations. A linear equation is in the form:

\[a_1x_1 + a_2x_2 + a_3x_3 + \ldots + a_nx_n = b \nonumber \]

Where \(a_1,a_2,a_3, \ldots a_n\) and \(b\) are known constants and \(x_1,x_2,x_3, \ldots x_n \) are unknown values. Typically we have systems of equations with different values of \(a\)s and \(b\)s but the unknowns are the same. For example. Consider the example of linear equations in the following video.

from IPython.display import YouTubeVideo
YouTubeVideo("CH68cc7sH4A",width=640,height=360, cc_load_policy=True)

Giselle works as a carpenter and as a blacksmith. She earns 20 dollars per hour as a carpenter and 25 dollars per hour as a blacksmith. Last week, Giselle worked both jobs for a total of 30 hours, and earned a total of 690 dollars. How long did Giselle work as a carpenter last week, and how long did she work as a blacksmith?

• Brainly.com

This problems gives us two equations and two unknowns:

\[ c + b = 30 \nonumber \]

\[ 20c + 25b = 690 \nonumber \]

How would we solve this in linear algebra?

\[ c + b = 30 \nonumber \]

\[ 20c + 25b = 690 \nonumber \]

First, we can multiply the first equation by -20 and add to the second equation. This is often called a “linear combination” of the two equations. The operation does not change the answer:

\[ -20c - 20b = -600 \nonumber \]

\[ 20c + 25b = 690 \nonumber \]

\[---- \nonumber \]

\[ 0c + 5b = 90 \nonumber \]

This is our new system of equations: \(c+b=300c+5b=90\)

Now we can easily divide the second equation by 5 and get the value for \(b\):

\[b = 90/5 = 18 \nonumber \]

If we substitute 18 for \(b\) into the first equation we get: \(c+18=30\)

And solving for \(c\) gives us \(c\)=30−18=12. Let’s check to see if this works by substituting \(c\)=18 and \(c\)=12 into our original equations:

\[ 12 + 18 = 30 \nonumber \]

\[ 20(12) + 25(18) = 690 \nonumber \]

Let’s check the answer using Python:

b = 18
c = 12

c + b == 30

True

20*c + 25*b == 690

True