# Solving Problems By Elimination

Multiplication can be used to set up matching terms in equations before they are combined.When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate.

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Example 3: $$\begin 2x - 5y &= 11 \\ 3x 2y &= 7 \end$$ Solution: In this example, we will multiply the first row by -3 and the second row by 2; then we will add down as before.

$$\begin &2x - 5y = 11 \color\\ &\underline \end\\ \begin &\underline} \text\\ &19y = -19 \end$$ Now we can find: back into first equation: $$\begin 2x - 5\color &= 11 \\ 2x - 5\cdot\color &= 11\\ 2x 5 &= 11\\ \color &\color \color \end$$ The solution is $(x, y) = (3, -1)$.

Recall that a false statement means that there is no solution.

If both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. A theater sold 800 tickets for Friday night’s performance. Combining equations is a powerful tool for solving a system of equations.

$$3y 2x=6$$ $$\underline$$ $$=8y\: \: \: \: \; \; \; \; =16$$ $$\begin \: \: \: y\: \: \: \: \: \; \; \; \; \; =2 \end$$ The value of y can now be substituted into either of the original equations to find the value of x $$3y 2x=6$$ $$3\cdot 2x=6$$ $$6 2x=6$$ $$x=0$$ The solution of the linear system is (0, 2).

To avoid errors make sure that all like terms and equal signs are in the same columns before beginning the elimination.

The elimination method of solving systems of equations is also called the addition method.

To solve a system of equations by elimination we transform the system such that one variable "cancels out".

Just as with the substitution method, the elimination method will sometimes eliminate both variables, and you end up with either a true statement or a false statement.

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