Elimination Method: Easy Steps for 8th Graders

The elimination method is a powerful technique for solving systems of linear equations.When choosing between elimination and substitution methods, we need to look at the coefficients of our variables.In elimination, we look for variables that have coefficients that are either opposites or can be made into opposites.Proper alignment is crucial for the elimination method. We align like terms vertically - x terms with x terms, y terms with y terms.Each column represents a different type of term - our x terms, y terms, and constants.Here's an ideal case for elimination - notice how the y terms are already opposites.And here's a case where elimination might not be the best first choice - no terms are opposites or easily made opposite.Let's review when elimination is the best choice for solving systems of equations.Elimination works best when coefficients are opposites, can easily be made opposite, or when no variable has a coefficient of one.When solving systems of equations using elimination, we often need to modify our equations to create opposite coefficients.Here, we have 2x in the first equation and 3x in the second equation. These coefficients aren't opposites yet.To make them opposite, we'll multiply the first equation by 3.And multiply the second equation by negative 2.Now let's look at what happened to our x terms.Let's look at another example with different coefficients.In this case, we have 4x and 5x. To make them opposite, we need to find numbers that will give us equal coefficients with opposite signs.We multiply the first equation by 5 and the second by negative 4.Here are some helpful tips for finding the right multipliers.Now that we have equations with opposite coefficients, we can add them to eliminate x.First, let's align our equations carefully, making sure like terms are in vertical columns.When we add these equations, the x terms will cancel out because six x minus six x equals zero.Adding the y terms, we get two y plus three y equals five y. The right side becomes sixteen minus nine, which is seven.Now we can solve this simpler equation for y. Dividing both sides by five gives us y equals seven fifths.To find x, we substitute y equals seven fifths into one of our original equations. Let's use six x plus two y equals sixteen.Simplifying the substitution, two times seven fifths becomes fourteen fifths.Moving all terms with x to the left side and all other terms to the right side.Converting sixteen to eighty fifths and subtracting fourteen fifths.This simplifies to sixty-six fifths.Finally, dividing both sides by six gives us x equals eleven fifths.