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Algebra
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13 Nov 2019

Solve the following pairs of simultaneous equations algebraically (Can you try to show your working in detail and simple to understand because I multiplied the first equation by 2 and the second equation by 3; I started getting fraction, which I found hard)

3x-4y=6, 2x+3y=8

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Bunny Greenfelder
Bunny GreenfelderLv2
11 Mar 2019

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Show transcribed image text

1)

x - y + 2z = 0

3x + 4y + az = 0

2x + by + 4z = 0

show that the system of equations has a non-trivialsolution either a = 6 or b = -2.

Find the general solution to the system of equations whena = 6.


2)

Let A =

7 -8 0

4 -5 0

-1 1 1

If possible, find a diagonal matrix C and an invertiblematrix B such that A = BCB-1

I have found the characteristic polynomial of A, alongwith the eigenvalues/vectors. However I just don't understand thisquestion, and my notes/textbook is completely useless! I assume thetwo matrices A and C will be similar, thus have the sameeigenvalues. But how am I supposed to show that matrix!?


3)

Consider you have a 3x3 matrix like this:

a b 0

0 a b

0 0 a

**the text is what is contained in the picture**

1. Consider first a linear equation: 3x – 1 = 5.

a. Use a graphing utility to show y1= 2x – 1 and y2=5 and determine the point of intersection. Make a sketch and label it.

b. How does the graph in part (a) relate to the solution set of the equation 3x – 1 = 5?

c. To solve the equation 3x – 1 =5 algebraically, the first step is to add 1 to both sides of the equation. Use a graphing utility to show y1=3x and y2=6, and determine the point of intersection. Make a sketch and label it. How does this graph relate to the equation 3x=6?

d. The final step to solve the equation is to divide both sides of the equation by 3. Use a graphing utility to show y1=x and y2=2, and determine the point of intersection. Make a sketch and label it. How does this graph relate to the equation x=2?

e. The algebraic steps to solve the equation 3x – 1 = 5 produce two equations: 3x=6 and x=2. How do the graphs you sketched above represent the fact that these equations are equivalent to the original?

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