How to Solve Special Systems in Algebra

Special systems have no solution or an infinite number of solutions.
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A special system consists of two linear equations that are parallel or have an infinite number of solutions.To solve these equations, you add or subtract them and solve for the variables x and y. Special systems may seem challenging at first, but once you practice these steps, you'll be able to solve or graph any similar type of problem.

No Solution

    Write the special system of equations in a stack format. For example: x+y=3 y= -x-1.

    Rewrite so the equations are stacked above their corresponding variables.

    y= -x +3 y= -x-1

    Eliminate the variable(s) by subtracting the bottom equation from the top equation. The result is: 0=0+4. 0≠4. Therefore, this system has no solution. If you graph the equations on paper, you will see that the equations are parallel lines and do not intersect.

Infinite Solution

    Write the system of equations in a stack format. For example: -9x -3y= -18 3x+y=6

    Multiply the bottom equation by 3: \=3(3x+y)=3(6) \=9x+3y=18

    Rewrite the equations in stacked format: -9x -3y= -18 9x+3y=18

    Add the equations together. The result is: 0=0, which means that both equations are equal to the same line, thus there are infinite solutions. Test this by graphing both equations.

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