Elimination by Addition
Solve systems of equations using elimination by addition in Grade 9 algebra. Add equations to cancel a variable, then solve the resulting single-variable equation.
Key Concepts
Property When two linear equations have a variable with opposite coefficients (e.g., $ax$ and $ ax$), add the equations to eliminate that variable. Explanation Think of this as a mathematical disappearing act! When one variable in an equation is the exact opposite of its counterpart in the other equation, like a hero and its anti hero ($+5x$ and $ 5x$), you can simply add the two equations. The opposite terms cancel each other out, leaving a much simpler problem to solve. Examples Given $3x + 4y = 10$ and $ 3x + 2y = 8$, adding them yields $6y = 18$, so $y = 3$. For the system $x + y = 5$ and $ x + y = 1$, adding the equations results in $2y = 6$, so $y = 3$. In the system $8a 5b = 11$ and $ 8a 3b = 5$, adding them gives $ 8b = 16$, which simplifies to $b = 2$.
Common Questions
How does elimination by addition work for systems of equations?
Add the two equations so one variable cancels (coefficients sum to zero). Solve the resulting one-variable equation, then substitute back to find the other variable.
When do you need to multiply before using elimination?
Multiply one or both equations by constants when the variable you want to eliminate does not already have opposite coefficients. Make them opposite so they cancel when added.
What does the solution to a system represent geometrically?
The solution is the intersection point of both lines on a coordinate plane. Elimination finds the exact x and y coordinates of that point.