Systems of Linear Equations

Systems of Linear Equations are comprised of several linear equations that share the same variables. The solution is found at the point(s) where all the equations intersect. A variety of methods such as substitution, elimination, graphical representation, or matrix utilization can be used to solve them. These equations have significant applications in diverse fields like physics, engineering, economics, and computer graphics.

Substitution Method

Solve the following system of equations using the substitution method:

\begin{aligned} 3 x + 2 y & = 12, \\ x - y & = 2. \end{aligned}

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Determining Parallel Lines

Given the linear equations

2 x + 3 y = 6

and

4 x - k y = 12

, for what value of

k

will the lines be parallel?

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Determining Perpendicular Lines

Given the equation of a line as

y = 3 x + 2

, find the equation of the line that is perpendicular to it and passes through the point (2, -1).

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Cramer's Rule

Solve the following system of equations using Cramer's Rule:

\begin{aligned} 2 x + 3 y & = 7 \\ 4 x - y & = 1 \end{aligned}

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Solving using Matrices by Elimination

Solve the following system of equations using matrix elimination method:

3 x - 2 y = 7

and

5 x + y = 11

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Solving using Matrices by Row Operations

Solve the following system of linear equations using matrices by row operations:

\begin{aligned} 2 x - 3 y + z & = 9 - x + 4 y - z & = - 7 3 x - 2 y + 2 z & = 12 \end{aligned}

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Solving using an Augmented Matrix

Solve the following system of linear equations using an augmented matrix:

\begin{aligned} 3 x - 2 y + z & = 1, \\ 2 x + y - z & = - 1, \\ x + 2 y + 3 z & = 5. \end{aligned}

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Systems of Linear Equations

Substitution Method

Determining Parallel Lines

Determining Perpendicular Lines

Cramer's Rule

Solving using Matrices by Elimination

Solving using Matrices by Row Operations

Solving using an Augmented Matrix

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