Lineaire functies en vergelijkingen *: Eerstegraadsvergelijkingen oplossing *
Solving by reduction
We discuss a systematic approach to solving linear equations with a single unknown.
Reduction
A linear equation with a single unknown can be solved by reduction: performing a series of algebraic operations, each time the same operation on both the left and right hand side, so as to simplify the equation.
Examples are
- subtracting a term from both sides of the equation
- dividing both sides of the equation by the same nonzero constant
Like terms can be collected at all times.
We carry these operations out so as to reduce an equation like #2x+5=4x-7# to an equation of the form #x=6#, for then we will have found that the solution #x# has value #6#.
In other words, a linear equation with unknown #x# can be solved by moving all of the terms containing #x# to the left, moving all of the constants to the right, collecting terms, and dividing both sides by the coefficient of #x#.
If all terms with #x# are collected and the coefficient of #x# is equal to #0#, then #x# has disappeared from the equation. Later, in theory The general solution, we describe what happens then.
The idea behind reduction is that we know in advance that the solutions to the original equations are exactly the same as the solutions to the new equation. In such a case we call the original equation and the reduced equation equivalent.
Equivalence of equations Two equations are called equivalent if they have the same solutions.
Each of the two equations #x^6=0# and #x^2=0# is equivalent to #x=0#.
The equations #y^6=-1# and #\frac{1}{y}=0# are equivalent: they both have no solutions.
The following steps can be followed to solve the equation.
\[\begin{array}{rclcl} 7x + 3 &=& 9x -1 &\phantom{x}&\color{blue}{\text{the given equation}}\\7x + 3 - 9x &=& 9x -1 - 9x &\phantom{x}&\color{blue}{9x\text{ subtracted}}\\-2x +3 &=& -1 &\phantom{x}&\color{blue}{\text{simplified}}\\-2x +3-3 &=& -1 -3&\phantom{x}&\color{blue}{\text{constant } 3\text{ subtracted}}\\-2x &=& -4&\phantom{x}&\color{blue}{\text{simplified}}\\ x &=& \frac{-4}{-2}&\phantom{x}&\color{blue}{\text{divided by }-2}\\ x &=& {2}&\phantom{x}&\color{blue}{\text{fraction simplified}}\end{array}\]Therefore, the solution is #x ={ 2}#.
Or visit omptest.org if jou are taking an OMPT exam.
