Example Card: Elimination Using Multiplication
Solve systems of equations by multiplying to create opposite coefficients, then eliminating a variable. Practice elimination method for Grade 9 linear systems.
Key Concepts
Sometimes coefficients aren't ready for elimination. Here, we'll use the second key idea: multiplying equations to create an opening.
Example Problem Solve the system by elimination: $3x + 4y = 16$ and $5x 6y = 30$.
Step by Step 1. To get opposite coefficients for one variable, we can multiply the first equation by $3$ and the second equation by $2$. This will make the $y$ coefficients $ 12$ and $12$. $$ \begin{aligned} 3(3x + 4y = 16) & \rightarrow & 9x + 12y = 48 \\ 2(5x 6y = 30) & \rightarrow & 10x 12y = 60 \end{aligned} $$ 2. Now, add the two new equations to eliminate the $y$ variable. $$ \begin{array}{r} 9x + 12y = 48 \\ 10x 12y = 60 \\ \hline 19x \quad = 108 \end{array} $$ 3. Solve for $x$. $$ x = \frac{108}{19} $$ 4. Substitute $\frac{108}{19}$ for $x$ in the first original equation to find $y$. $$ \begin{aligned} 3(\frac{108}{19}) + 4y &= 16 \\ \frac{324}{19} + 4y &= 16 \\ 4y &= 16 \frac{324}{19} \\ 4y &= \frac{304}{19} \frac{324}{19} \\ 4y &= \frac{20}{19} \\ y &= \frac{5}{19} \end{aligned} $$ 5. The solution is $(\frac{108}{19}, \frac{5}{19})$.
Common Questions
When do you need to multiply before using elimination?
When neither variable has opposite coefficients. Multiply one or both equations by constants to create coefficients that cancel when added together.
How do you choose what to multiply in the elimination method?
Find the LCM of coefficients for one variable. For 4y and 6y, LCM is 12: multiply equation 1 by 3 and equation 2 by 2 to get 12y and -12y.
What do you do after eliminating one variable?
Solve for the remaining variable, then substitute that value back into either original equation. Check your answer in both equations to verify.