Q: In some circumstances, your system of linear equations may have infinite solutions. This means that any pair of values that you insert into the two variables will make the two equations correct. This happens when the two equations are really just algebraic variations of the same, single equation.  For example, consider these two equations:  2x+8y=18{\displaystyle 2x+8y=18} x+4y=9{\displaystyle x+4y=9}   If you begin working on this system and try creating a pair of matching coefficients, you will find that by multiplying the second equation by 2 you will create the equation 2x+8y=18{\displaystyle 2x+8y=18}. This is an exact match of the first equation. If you proceed through the steps, you will eventually get the result 0=0{\displaystyle 0=0}. A solution of 0=0 means that you have “infinite” solutions or you can simply say that the two equations are identical. If you consider this system graphically and plot the lines that are represented by the two equations, the “infinite” solution means that the two lines lie exactly one on top of the other. It is really only one line. Occasionally you may have a system in which the two equations, when written in standard form, are nearly identical except that the constant term C is different. Such a system has no solution.  Consider these equations:  4x+2y=6{\displaystyle 4x+2y=6} 2x+y=4{\displaystyle 2x+y=4}   At first glance, these look like very different equations. However, when you begin solving and multiply each term of the second equation by 2 to try to create matching coefficients, you will wind up with the two equations:  4x+2y=6{\displaystyle 4x+2y=6} 4x+2y=8{\displaystyle 4x+2y=8}   This is an impossible situation, since the expression 4x+2y{\displaystyle 4x+2y} cannot equal both 6 and 8 at the same time. If you were to try solving this by subtracting the terms, you would reach the result 0=−2{\displaystyle 0=-2}, which is an incorrect statement. In such a circumstance, your response is that there is no solution to this system. If you consider what this system means graphically, these are two parallel lines. They will never intersect, so there is no single solution to the system. It is possible for a system of linear equations to have more than two variables. You may have 3, 4, or as many variables as the problem dictates. Finding a solution to the system means finding a single value for each variable that makes each equation in the system correct. To find a single, unique solution, you must have as many equations as you have variables. Thus, if you have the variables x,y{\displaystyle x,y} and z{\displaystyle z}, you need three equations. Solving a system of three or more variables can be done using the linear combinations explained here, but that gets very complicated. The preferred method is using matrices, which is too advanced for this article. You may wish to read Use a Graphing Calculator to Solve a System of Equations.
A: Recognize identical equations as having infinite solutions. Find systems with no solution. Use a matrix for systems with more than two variables.

Q: Fill a bowl with 1 cup (240 mL) of water. Add the juice from half a lemon and 1 tbsp (8 g) of baking soda. Stir to create a soaking solution. Soak your nails for 15-20 minutes, then rinse with fresh water.  Lemon is a natural stain remover and can remove yellow stains. It's often used as a whitener! Baking soda is also a stain remover. Use a high-grit buffer, such as a 220 or greater. Lightly rub the buffer over the surface of the gel until the dingy layer is gone. You should see the original color of the gel return. This will protect the manicure and help prevent it from getting dingy again. Lightly brush a thin layer of top coat over your nails, then let it completely dry.  You can use your normal top coat over the gel polish. Continue to add a layer of topcoat every few days to keep your manicure looking fresh!
A:
Soak your nails in lemon juice and baking soda to remove discoloration. Buff the top of your gel manicure to remove the dingy layer. Apply a layer of topcoat over your manicure after you buff it.