### Systems of linear equations and matrices: Linear equations

### Reduction to a base form

Here we discuss a systematic method of reducing a linear equation to a basic form.

**Reduction** Reduction of a linear equation is a process of step-by-step simplifying the equation by

- applying the same elementary operation to the left hand side and right hand side of the equation (for example: subtracting the same term from both sides or dividing by the same constant distinct from \(0\))
- collecting terms
- expanding brackets

We carry out these operations with the intention to arrive at a simpler equation, like a *base form* \[a_1\cdot x_1 + \cdots + a_n\cdot x_n + b = 0\] with unknowns #x_1,\ldots,x_n# and numbers #a_1,\ldots,a_n,b#.

In other words, a linear equation can be reduced by expanding brackets, bringing all the terms with an unknown to the left, bringing all constants to the left, collecting similar terms, and, if desired, dividing both sides by a constant equal to zero.

The examples illustrate the systematic approach of reduction.

A possible basic form: \(2x-2y+1=0\)

This follows from the following reduction.

\[\begin{array}{rclcl} 5x+5y+6&=& 3x+7y+5 &\phantom{x}&\color{blue}{\text{the given equation}}\\5x+5y+6- 3x &=& 3x+7y+5 - 3x &\phantom{x}&\color{blue}{3x\text{ subtracted on both sides}}\\2x+5y+6 &=& 7y+5 &\phantom{x}&\color{blue}{\text{simplified}}\\ 2x+5y+6- 7y &=& 7y+5 - 7y &\phantom{x}&\color{blue}{7y\text{ subtracted on both sides}}\\2x-2y+6 &=& 5 &\phantom{x}&\color{blue}{\text{simplified}}\\2x -2y+6-5 &=& 5 -5&\phantom{x}&\color{blue}{ 5\text{ subtracted on both sides}}\\2x-2y+1 &=& 0&\phantom{x}&\color{blue}{\text{simplified}}\end{array}\] So a basic form of the linear equation is \(2x-2y+1=0\).

This follows from the following reduction.

\[\begin{array}{rclcl} 5x+5y+6&=& 3x+7y+5 &\phantom{x}&\color{blue}{\text{the given equation}}\\5x+5y+6- 3x &=& 3x+7y+5 - 3x &\phantom{x}&\color{blue}{3x\text{ subtracted on both sides}}\\2x+5y+6 &=& 7y+5 &\phantom{x}&\color{blue}{\text{simplified}}\\ 2x+5y+6- 7y &=& 7y+5 - 7y &\phantom{x}&\color{blue}{7y\text{ subtracted on both sides}}\\2x-2y+6 &=& 5 &\phantom{x}&\color{blue}{\text{simplified}}\\2x -2y+6-5 &=& 5 -5&\phantom{x}&\color{blue}{ 5\text{ subtracted on both sides}}\\2x-2y+1 &=& 0&\phantom{x}&\color{blue}{\text{simplified}}\end{array}\] So a basic form of the linear equation is \(2x-2y+1=0\).

For example, another basic form can be obtained by dividing both sides of the newly acquired base by \( 2\) share, resulting in \(x-y+{{1}\over{2}}=0\).

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