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SAT Maths: System Of Linear Equations

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06/07/2017 0 0

Linear Equations - System Of Equations

A system of linear equations can be represented by lines on the coordinate plane.

Unique Solution / Intersecting Lines:

Graphically, a pair of linear equations having a Unique solution would be two intersecting lines on the coordinate plane. In such a case, the pair would be such that the ratio of the coefficients of x is not equal to that of y.

Example 1

2x + 3y = 5 and 5x + 6y = 11 are two linear equations that form a system of equations.

2/5 is 0.4 and is not equal to 3/6 which is 0.5. Hence the above equations have a unique solution.

No Solution / Parallel Lines:

Let ax+by+c = 0 and px+qy+r=0 are two linear equations in x and y with a, b, p, q being constants. If (a/p), which is the ratio of coefficients of x = (b/q), which is the ratio of coefficients of y is not equal to (c/r), the ratio of the constants, then the pair of linear equations has No solution. On the coordinate plane, they would be represented by parallel lines.

Example 2:

2x + 3y = 5 and 4 x + 6y = 11

2/4 which is ratio of coefficients of x is equal to 3/6 which is the ratio of coefficient of y but not equal to the ratio of the constants 5/11. Hence the above equations have no solution.

Infinite Solutions / Coinciding Lines:

A pair of linear equations has infinitely many solutions when (a/p), the ratio of the coefficients of x is equal to (b/q), the ratio of the coefficients of y is equal to (c/r), the ratio of the constants. On the coordinate plane, they would be represented by two lines one over another.

Example 3:

2x + 3y = 5 and 4x + 6y = 10

2/4 is equal to 3/6 is equal to 5/10. Hence, the above pair of linear equations has infinite solutions.

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