Multiplying to Create Opposites
Use the elimination method in Grade 9 algebra by multiplying equations by constants to create opposite coefficients, then adding equations to cancel one variable and solve the system.
Key Concepts
Property If no variables have the same or opposite coefficients, multiply one or both equations by a non zero constant to create a pair of opposite coefficients. Explanation Sometimes, your equations aren't ready to cooperate and don't have matching or opposite terms. This is where you become the director! You can multiply one or even both equations by a strategic number to create the perfect set of opposites. Once you've engineered the cancellation, you just add the equations together and solve away. Examples System: $3x + 2y = 7$ and $6x 5y = 4$. Multiply the first equation by $ 2$ to get $ 6x 4y = 14$ and eliminate $x$. System: $3a + 2b = 8$ and $2a 5b = 1$. Multiply the first by $2$ and the second by $ 3$ to get $6a+4b=16$ and $ 6a+15b=3$.
Common Questions
What does 'multiplying to create opposites' mean in solving systems?
In the elimination method, multiply one or both equations by constants so that the coefficients of one variable become additive inverses (opposites). Adding the equations then eliminates that variable entirely.
How do you eliminate a variable by multiplying to create opposites?
For the system 2x + 3y = 12 and 5x + y = 7: multiply the second equation by -3 to get -15x - 3y = -21. Add to the first: (2x - 15x) + (3y - 3y) = 12 - 21, giving -13x = -9 and x = 9/13.
When do you need to multiply both equations, not just one?
Multiply both equations when no single multiplier creates opposites for either variable. For 3x + 2y = 8 and 5x + 3y = 13: multiply first by 3 (9x + 6y = 24) and second by -2 (-10x - 6y = -26) to eliminate y.