The Step-by-Step Method (Multiplying First)
The step-by-step elimination method with multiplication is a Grade 7 algebra skill in Big Ideas Math Advanced 2, Chapter 5: Systems of Linear Equations. When solving systems by elimination, one or both equations may need to be multiplied by a constant to create opposite coefficients before adding. For example, to eliminate y in 3x plus 4y equals 11 and 5x minus 2y equals 13, multiply the second equation by 2 to get opposite y-coefficients, then add.
Key Concepts
Property When solving systems by elimination, you must write both equations in standard form first and clear any fractions. Sometimes, you must multiply one or both equations by a constant to make the coefficients of one variable opposites before adding.
Examples Multiplying One Equation: For the system $3x + 4y = 11$ and $5x 2y = 13$, multiply the second equation by 2 to get opposite $y$ coefficients. This results in $3x + 4y = 11$ and $10x 4y = 26$. Multiplying Both Equations: For the system $2x + 5y = 8$ and $3x + 7y = 12$, multiply the first by 3 and the second by 2 to get opposite $x$ coefficients. This results in $6x + 15y = 24$ and $ 6x 14y = 24$. Full Process: Solve $2x + 5y = 34$ and $ 3x + 2y = 44$. Multiply to make the x terms opposites: $3(2x + 5y) = 3( 34)$ and $2( 3x + 2y) = 2( 44)$. This simplifies to $6x + 15y = 102$ and $ 6x + 4y = 88$. Adding them gives $19y = 190$, so $y = 10$. Substitute back to get $x = 8$, making the solution $(8, 10)$.
Explanation Strategic multiplication creates the opposite coefficients needed for elimination to work effectively. Look for the variable that will be easiest to eliminate by examining both coefficients and choosing appropriate multipliers. Once you add the equations and solve for the first variable, always substitute that answer back into an original equation to find the second variable, and check your final ordered pair.
Common Questions
How do you use multiplication in the elimination method?
Multiply one or both equations by constants to make the coefficients of one variable equal and opposite. Then add the equations to eliminate that variable and solve for the remaining one.
When do you need to multiply both equations in elimination?
When neither variable has coefficients that are already opposites or equal, multiply both equations by different constants to create opposite coefficients for one variable.
What is the full elimination process for a system?
Step 1: Multiply equations to create opposite coefficients. Step 2: Add the equations to eliminate one variable. Step 3: Solve for the remaining variable. Step 4: Substitute back to find the other variable.
What textbook covers the elimination method with multiplication in Grade 7?
Big Ideas Math Advanced 2, Chapter 5: Systems of Linear Equations covers the elimination method including the step-by-step multiplying strategy.