Solving systems of equations by elimination

Systems of linear equations: Two equations with two unknowns

Solving systems of equations by elimination

We have seen how we can solve a system of equations by means of substitution. There is another method to solve systems of linear equations, namely the elimination method. This method can also be applied to systems with multiple linear equations with multiple unknowns, whereas using substitution can be quite complicated in the case of such systems.

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Elimination method

	Procedure	Example
	With the elimination method for solving a system of two linear equations with two unknowns, we proceed as follows.	Solve the following system: #\lineqs{\quad 2 \cdot x +4 \cdot y+5&= \quad 0 \cr \quad -3 \cdot x +2 \cdot y -4&= \quad 0 \cr}#
Step 1	Multiply the first and/or second equation with a number such that the coefficient of #x# or of #y# of both equations become equal.	#\lineqs{2 \cdot x +4 \cdot y+5=0 \cr -6 \cdot x +4 \cdot y -8=0 \cr}#
Step 2	Subtract the second equation from the first. We are now left with an equation with only #x# or #y#.	#\begin{array}{rcl}2 \cdot x +4 \cdot y+5&=&0 \\ -6 \cdot x +4 \cdot y -8&=&0\\ \hline 8\cdot x+13&=&0\end{array}-#
Step 3	Solve the linear equation of step 2 using reduction.	#x=-\frac{13}{8}#
Step 4	Substitute the value obtained in step 3 in one of the original equations to determine the value of the remaining unknown.	#\begin{array}{rcl}2 \cdot -\frac{13}{8}+4 \cdot y +5&=&0\\ 4 \cdot y +\frac{7}{4}&=&0\\ 4 \cdot y&=&-\frac{7}{4} \\ y&=&-\frac{7}{16} \end{array}#
Step 5	Give the answer in the form \[\lineqs{ x & =\;\; \ldots \\ y &=\;\; \ldots }\]	#\lineqs{ x &= \;\; -\frac{13}{8} \\ y &= \;\; -\frac{7}{16} }#

Further elimination

In step 4, it's also possible to perform step 1 through 3 again, but by multiplying the original equations so that the remaining unknown has the same coefficient in both equations.

Solve the following system of equations by using the elimination method.
\[\lineqs{6\cdot x+9\cdot y&=&2\cr
-8\cdot x-9\cdot y&=&-7\cr }\]

#\lineqs{x&=&{{5}\over{2}}\cr y&=&-{{13}\over{9}}\cr }#

In the system, the coefficient of #y# in the first equation is equal to minus one times the coefficient of #y# in the second equation. We can therefore eliminate the variable #y# from the first equation by replacing this equation with the sum of the given equations. Hence, we start with step 2 of the procedure.

Step 2	\[\begin{array}{rcl}6\cdot x+9\cdot y&=&2 \\ -8\cdot x-9\cdot y&=&-7\\ \hline -2\cdot x&=&-5\end{array}+\]
Step 3	We can reduce this equation in order to find #x#. This goes as follows: \[\begin{array}{rcl} -2\cdot x&=&-5 \\&&\phantom{x}\blue{\text{the equation we need to solve}} \\ x&=& {{5}\over{2}} \\&&\phantom{x}\blue{\text{both sides divided by }-2} \end{array}\]
Step 4	We substitute #x={{5}\over{2}}# in the second original equation. We subsequently solve this by reduction. This goes as follows: \[\begin{array}{rcl} -8\cdot {{{5}\over{2}}}-9\cdot y &=&-7 \\&&\phantom{xyz}\blue{\text{substituted }x={{5}\over{2}} \text{ in the second equation}}\\ -20-9\cdot y&=&-7\\&&\phantom{xyz}\blue{\text{calculated}}\\ -9\cdot y&=&13\\&&\phantom{xyz}\blue{\text{added }20\text{ on both sides}}\\ y&=&-{{13}\over{9}}\\&&\phantom{xyz}\blue{\text{both sides divided by }-9} \end{array}\]
Step 5	Hence, the solution to the system is \[\lineqs{x&=&{{5}\over{2}}\cr y&=&-{{13}\over{9}}\cr }\]

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