Solving equations and inequalities: Linear equations in one unknown
General solution rules
In general, the solutions of #ax+b=0# with unknown #x# be found as follows.
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\(\;\)case
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\(\;\)solution
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\(\;a\ne0\;\)
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\(\;\)exactly one: \(x=−\frac{b}{a}\;\)
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\(\;a=0\) and \(b\ne0\;\)
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\(\;\)no real solution\(\;\)
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\(\;a=0\) and \(b=0\;\)
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\(\;\)any real number \(x\;\)
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We explain why. The equation is \(ax+b=0\).
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\(\;\)case
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\(\;\)solutions
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\(\;\)explanation
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\(\;a\ne0\)
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\(\;\)exactly one: \(\;x=−\frac{b}{a}\;\)
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\(\;\)Subtract \(b\) the left and right-hand side, |
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\(\;a=0\) and \(b\ne0\)
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\(\;\)no solution
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\(\;\)The equation becomes \(b=0\) and this is
\(\;\)not true, irrespective of the choice of \(x\). |
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\(\;a=0\) and \(b=0\)
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\(\;\)any real number \(x\)
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\(\;\)The equation becomes \(0=0\)
\(\;\)and this is true for any value of \(x\) |
You do not need to memorise these rules, because the solutions can easily be found by reduction.
The three cases can also be recognised in terms of straight line (i.c. intersecting, parallel, identical)
We will later discuss examples of more general equations that can be reduced to a linear equation.
Mathcentre video
Solving Linear Equations (34:00)
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