Help to solve a linear equation

In summary, Sandra posted on a forum seeking help with a math problem she and her boyfriend had given each other. She needed to use linear algebra to solve the problem, but had only read moderately on the topic. Another user responded by expanding the determinant using the first column and giving a hint for solving the resulting cubic equation. Sandra thanked the user and jokingly said she would give her boyfriend a poke in the eye.
  • #1
Sandra_xx
2
0
Hi!

This is my first post at this forum. At the look of it I think i will be spending a lot of time here. I´ve have a problem that i need help with. Me and my boyfriend started a discussion yeasterday who was the smartest of us. So we gave each other a assignment. Mine was:

Solve the equation

| 1 1 1 1|
| x 2 3 4|
| x^2 4 9 16| = 0
| x^3 8 27 64|

I´ve have thought out that i need to use linear algebra to solve this. This is were my problem kick in. I´ve have just read moderate linaer algebra. So please can you help me. Don´t want to be the dumb one o:) We didn´t say how we should solve it...

Thanks in advance! /X Sandra
 
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  • #2
Expand the determinant using the first column:
[tex]\left|\begin{array}{cccc}1 & 1 & 1 & 1 \\ x & 2 & 3 & 4 \\ x^2 & 4 & 9 & 16 \\ x^3 & 8 & 27 & 65 \end{array}\right|= [/tex]
[tex]\left|\begin{array}{ccc}2 & 3 & 4 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|[/tex]
[tex]- x \left|\begin{array}{ccc}1 & 1 & 1 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|[/tex]
[tex]+ x^2 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 8 & 27 & 64\end{array}\right|[/tex]
[tex]- x^3 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 4 & 9 & 16\end{array}\right|[/tex]

Which gives a single cubic (not linear) equation to solve. (Hint: it has one integer solution.)
 
  • #3
Got it now. Thank you :!)

X/ Sandra
 
  • #4
And give your boyfriend a poke in the eye with a sharp stick!
 

Related to Help to solve a linear equation

1. What is a linear equation?

A linear equation is an algebraic equation that represents a straight line on a graph. It has the form y = mx + b, where m is the slope of the line and b is the y-intercept.

2. How can I solve a linear equation?

To solve a linear equation, you need to isolate the variable on one side of the equation by using inverse operations. This means undoing any addition, subtraction, multiplication, or division that is being done to the variable. Once the variable is isolated, you can solve for its value.

3. What is the importance of solving linear equations?

Solving linear equations is important in many fields of science and mathematics, as it allows us to find the relationships between different variables and make predictions about their behavior. It is also a foundational skill in algebra and is necessary for more complex mathematical concepts.

4. Are there any common mistakes to avoid when solving linear equations?

Yes, there are a few common mistakes to avoid when solving linear equations. One is forgetting to use the same operation on both sides of the equation, which can lead to an incorrect solution. Another is not distributing a coefficient when solving an equation with parentheses. It is also important to check your solution by plugging it back into the original equation to ensure it is correct.

5. Are there any tips for solving linear equations more efficiently?

Yes, there are a few tips for solving linear equations more efficiently. One is to simplify the equation first by combining like terms or using the distributive property. Another is to start by isolating the variable with the highest coefficient, as this will often result in simpler calculations. It is also helpful to double-check your work and use a calculator if needed to avoid simple arithmetic errors.

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