What Is the Definition of a Common Solution in College Algebra?

Finding the common solution can be done with pencil and paper.
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Finding a common solution between two, or less frequently, more equations, is a bedrock skill in college algebra. Sometimes a math student is faced with two or more equations. In college algebra, these equations have two variables, x and y. Both carry an unknown value, which means in both equations, x stands for one number, and y stands for another. These two equations intersect at one point, where x and y have the same values for both. Finding these (x,y) values is the definition of the common solution.

Systems of Equations

The easiest way to understand this concept is to use an example, for instance, the equations y = 2x and y = 3x + 1. Independently, these two equations each have a range of values, the y value changing depending on which x value you plug into the equation. Together, however, these two equations have one common solution. With two equations, you can use them and the variables inside them to find out where the two equations meet.

Finding Plot Points

The first way to find the values of x and y is to graph the two equations, which means that first, you find plot points. This entails plugging in various x values and seeing which y value is then arrived at. For example, when you plug the values 0,1,2,3 into each equation and find the y values for both, you get the results 0,2,4,6 for the first equation and 1,4,7,10 for the second. Combine each of these with the x coordinates, which always come first in plot points, to get (0,0), (1,2), (2,4) and (3,6) for the first equation. The second yields the coordinates (0,1), (1,4), (2,7) and (3,10). The solution you’ll see is (-1,-2).

Graphing With X and Y Axes

Use a graph with an x and a y axis. To plot each point in the first equation, find the x and y values of each coordinate and mark a dot there. This means counting horizontally the number of each x value, and vertically the number of each y value. Once you have four plot points for the first equation, draw a line between them. Do the same for the second equation, then draw a line between them as well. The intersection is the common solution. Sometimes this is not the most elegant result, however.

Solving Algebraically

Instead, you can solve algebraically, by substitution, an x value in for y. Since y = 2x, you can put 2x in the second equation in its place. You then have the equation 2x = 3x + 1. This becomes -x = 1, which means x = -1. When you plug this into the simpler equation, this means y = 2(-1) or y = -2.

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