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Welcome to our Math lesson on Systems of Linear Equations, this is the second lesson of our suite of math lessons covering the topic of Systems of Linear Equations. Methods for Solving Them., you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
As pointed out in the previous paragraph, solving a linear equation means finding an infinity of possible pairs of values that, when combined, give a straight line as a graph. From geometry, it is known that a line represents a set of points, where each point represents a pair of coordinates that if inserted in the place of the corresponding variables, make the equation true. Therefore, it is impossible to solve a single linear equation, as the number of possible solutions is infinite. The only thing we can do to calculate exact values for variables in such conditions is to simultaneously solve a pair of linear equations. In this way, we (perhaps) obtain a common pair of numbers as a solution, shown graphically by the intercept point of the two lines that represent each line involved.
For example, solving the system of linear equations
means finding a pair of values (one x and one y), which when inserted into the two equations of the system, give true results. Hence, the pair of values x = 1 and y = -1 is the only solution set of this system as
and
Remark! Often you will see the linear equations involved in a system written in the form ax + by = -c instead of the standard form ax + by + c = 0.
Check whether the number pairs (2, 0), (4, 3) and (5, 2) are solutions for the system of linear equations
The number pair (2, 0) is not a solution set for the given system of linear equations as it gives a true result only in the first equation but not in the second. Indeed, substituting x = 2 and y = 0 in each equation, yields
and
The number pair (4, 3) is not a solution set for the given system of linear equations as it gives a true result only in the second equation but not in the first. Indeed, substituting x = 4 and y = 3 in each equation, yields
and
As for the number pair (5, 2), it is a solution set for the given system of linear equations, as it gives a true result in both equations. Indeed, substituting x = 5 and y = 3 in each equation, yields
and
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