Elimination by Multiplying Both Equations
Grade 9 students in California Reveal Math Algebra 1 learn to eliminate a variable from a system of equations by multiplying both equations by different constants, then adding. When coefficients are not simple multiples of each other, find the Least Common Multiple of the two coefficients, then scale each equation so the target variable has coefficients that are equal in magnitude but opposite in sign. For example, to eliminate y from 3x+4y=10 and 5x+6y=14: LCM of 4 and 6 is 12. Multiply the first by 3 and the second by -2 to get 12y and -12y; adding gives -x=2, so x=-2, then back-substitute for y=4.
Key Concepts
Property When the coefficients of the variable you want to eliminate are not simple multiples of each other, you must multiply BOTH equations by different constants. Find the Least Common Multiple (LCM) of the two coefficients. Then, multiply each equation by a constant that turns that variable's coefficient into the LCM, ensuring one is positive and one is negative so they sum to zero.
Examples Example 1 (Eliminate y): Solve $3x + 4y = 10$ and $5x + 6y = 14$. The LCM of the $y$ coefficients (4 and 6) is 12. Multiply the first equation by $3$ to get $12y$, and the second by $ 2$ to get $ 12y$: $3(3x + 4y = 10) \rightarrow 9x + 12y = 30$ $ 2(5x + 6y = 14) \rightarrow 10x 12y = 28$ Add the new equations: $(9x 10x) + (12y 12y) = 30 28 \rightarrow x = 2 \rightarrow x = 2$. Example 2 (Back Substitution): Substitute $x = 2$ into the first original equation: $3( 2) + 4y = 10 \rightarrow 6 + 4y = 10 \rightarrow 4y = 16 \rightarrow y = 4$. The solution is $( 2, 4)$.
Explanation When neither number easily scales into the other, you have to find a common meeting point. This is exactly like finding a common denominator when adding fractions! If you have a 4 and a 6, they both comfortably meet at 12. You scale the top equation up to reach 12, and you scale the bottom equation to reach 12. Once they are perfect opposites, you add them together to trigger the elimination.
Common Questions
When do you need to multiply both equations in elimination?
When the coefficients of the variable you want to eliminate are not simple multiples of each other, you must multiply both equations by different constants to create opposite coefficients.
How do you choose what to multiply each equation by?
Find the LCM of the two coefficients. Multiply each equation by the value that turns its coefficient into the LCM, making one positive and one negative so they sum to zero.
How do you eliminate y from 3x+4y=10 and 5x+6y=14?
LCM of 4 and 6 is 12. Multiply first equation by 3: 9x+12y=30. Multiply second by -2: -10x-12y=-28. Add: -x=2, so x=-2.
How do you find y after eliminating one variable?
Substitute the found value back into one of the original equations. With x=-2 in 3x+4y=10: -6+4y=10, so 4y=16, y=4. The solution is (-2,4).
Why is finding the LCM like finding a common denominator?
Just as fractions need a common denominator to be added, systems need a common coefficient value for elimination. You scale each equation to reach that common value, making the terms cancel when added.
Which unit covers elimination by multiplying both equations?
This skill is from Unit 6: Systems of Linear Equations and Inequalities in California Reveal Math Algebra 1, Grade 9.