To solve systems of linear equations, there are several methods that you can use, including:
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Graphical Method: This involves graphing each equation on the same set of axes and identifying the point(s) where the lines intersect. This point of intersection represents the solution of the system.
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Substitution Method: You solve one of the equations for one variable and then substitute that expression into the other equation. This method is useful when one of the equations is easily solvable for one variable.
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Elimination Method: Also known as the addition method, this involves adding or subtracting the equations to eliminate one of the variables, making it possible to solve for the other variable.
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Matrix Method: If the system is large, you can use matrices and methods such as Gaussian elimination or using the inverse of a matrix.
Example:
Let's consider a simple system of linear equations:
- \(2x + 3y = 6\)
- \(x - y = 1\)
Solution using Substitution:
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From the second equation, solve for \(x\):
\[ x = y + 1 \]
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Substitute \(x\) in the first equation:
\[ 2(y + 1) + 3y = 6 \]
Simplifying this gives:
\[ 2y + 2 + 3y = 6 \ 5y + 2 = 6 \ 5y = 4 \ y = \frac{4}{5} \]
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Now substitute \(y\) back to find \(x\):
\[ x = \frac{4}{5} + 1 = \frac{9}{5} \]
Thus, the solution to the system of equations is:
\[ x = \frac{9}{5}, \quad y = \frac{4}{5} \]
If you have a specific system of equations in mind, please provide those equations, and I can help you solve them!