So far, we have learned how to solve a system of linear equations in two variables using graphing and substitution. Our graphing method involves graphing each equation and looking for the point of intersection. This point lies on both lines and is, therefore, a solution for
our system. We found that the graphing method was very inefficient and didn’t work well for non-integer values. We also learned how to use the substitution method. This method involves solving one of the equations of the system for one of the variables. We can then plug in for that variable in the other equation. This allows us to obtain a linear equation in one variable and find one of our unknowns. We can finish up the process by plugging in our known value into one of the equations and
solving for the final unknown. Show Elimination Method
Let's take a look at an example. Inconsistent Systems - Elimination Method At this point, we should know that there are linear systems that will not have a solution. This occurs when we have two parallel lines. Parallel lines have the same slope and will never intersect. This means there will never be a point that lies on both lines and there can't be a solution for the system. We saw with the substitution method, that solving
an inconsistent system or a system with no solution leads to a false statement. We will see the same result when we use the elimination method. A system with no solution will lead to the variables dropping out and a false statement. Let's look at an example. Dependent Equations - Elimination Method We have also seen a system with dependent equations. This means we have a system with the same two equations involved. The equations have been algebraically manipulated to look different, but they are the same. Since a linear equation in two variables has an infinite number of solutions, we can say there is an infinite number of solutions for this type of system. We will know we have
dependent equations when our variables drop out and we are left with a true statement. Let's look at an example. Skills Check:Example #1 Solve each system by elimination. $$7x - 7y=-7$$ $$-9x - 4y=22$$ Please choose the best answer. Example #2 Solve each system by elimination. $$-6x + 4y=0$$ $$-4x + 6y=20$$ Please choose the best answer. A $$\text{Infinite Solutions}$$ Example #3 Solve each system by elimination. $$11x + 7y=9$$ $$4x - 3y=31$$ Please choose the best answer. Congrats, Your Score is 100% Better Luck Next Time, Your Score is % Try again? What is the elimination method formula?The elimination method is the best choice of use when the equations are in the standard form Ax+By=C, and all the variables have coefficients other than 1.
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