Lesson 6-4: Solve Systems of Equations by Elimination

Property

If two pairs of expressions are equal, you can add or subtract the corresponding sides to create a new, valid equation. When coefficients of a variable have opposite signs, add the equations to eliminate that variable. When coefficients of a variable have the exact same sign, subtract the equations to eliminate the variable.

Examples

• Adding to Eliminate: Given the equations $x - 4y = 5$ and $2x + 4y = 10$, the coefficients of $y$ are $-4$ and $4$ (opposite signs). Adding the equations eliminates the $y$ term completely.
• Subtracting to Eliminate: Given the equations $3x + 2y = 8$ and $x + 2y = 4$, the coefficients of $y$ are both $2$ (same signs). Subtracting the second equation from the first gives $(3x + 2y) - (x + 2y) = 8 - 4$, which simplifies to $2x = 4$.

Explanation

Think of it as teaming up your equations to knock out a variable! The key to elimination is recognizing coefficient signs to determine the correct operation. When coefficients have opposite signs, adding the equations causes them to cancel out because a positive plus a negative equals zero. When coefficients have the same sign, subtracting eliminates the variable. It’s a clean knockout that simplifies the whole problem instantly!

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.

Property

When using elimination, if both variables are eliminated, the resulting mathematical statement determines the number of solutions:

• If the resulting equation is a false statement (e.g., $0 = c$), the system has no solution.
• If the resulting equation is a true statement (e.g., $0 = 0$), the system has infinitely many solutions.

You can always verify your algebraic solution by graphing the two lines to confirm they intersect exactly at your calculated $(x, y)$ coordinate.

Examples

• No Solution: For the system $2x - y = 5$ and $-4x + 2y = -12$, multiply the first equation by 2 to get $4x - 2y = 10$. Adding this to the second equation results in $0 = -2$, a false statement, meaning there is no solution.
• Infinitely Many Solutions: For the system $x + 3y = 6$ and $2x + 6y = 12$, multiply the first equation by -2 to get $-2x - 6y = -12$. Adding this to the second equation results in $0 = 0$, a true statement, meaning there are infinitely many solutions.
• Graphical Check: If elimination gives $(4, -1)$ as a solution, both original equations should pass exactly through the point $(4, -1)$ when graphed.

Explanation

Sometimes when you perform elimination, both variables cancel out. If you are left with a false statement like $0 = -2$, it means there is no pair of $(x, y)$ that can satisfy both equations because the lines are perfectly parallel. If you are left with a true statement like $0 = 0$, it means the two equations describe the exact same line, resulting in infinitely many solutions.