Lesson 4: Solve Systems by Elimination

Property

Step 1. Write both equations in standard form. If any coefficients are fractions, clear them.
Step 2. Make the coefficients of one variable opposites.

• Decide which variable you will eliminate.
• Multiply one or both equations so that the coefficients of that variable are opposites.

Step 3. Add the equations resulting from Step 2 to eliminate one variable.
Step 4. Solve for the remaining variable.
Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
Step 6. Write the solution as an ordered pair.
Step 7. Check that the ordered pair is a solution to both original equations.

Examples

• Solve $\begin{cases} 4x + 3y = 1 \\ x - 3y = 4 \end{cases}$. The equations are in standard form and the $y$ terms are opposites. Add them: $5x=5$, so $x=1$. Substitute into the second equation: $1-3y=4$, so $-3y=3$ and $y=-1$. The solution is $(1, -1)$.
• Solve $\begin{cases} x + \frac{1}{3}y = 2 \\ \frac{1}{2}x + \frac{1}{4}y = 2 \end{cases}$. First, clear fractions by multiplying the first equation by 3 and the second by 4 to get $\begin{cases} 3x+y=6 \\ 2x+y=8 \end{cases}$. Multiply the second equation by $-1$ and add: $(3x+y) + (-2x-y) = 6-8$, which gives $x=-2$. Then $3(-2)+y=6$, so $y=12$. The solution is $(-2, 12)$.
• Solve $\begin{cases} y = 5 - 2x \\ 3x - 2y = 4 \end{cases}$. First, rewrite the first equation in standard form: $2x+y=5$. Now multiply it by 2 to get $4x+2y=10$. Add this to $3x-2y=4$ to get $7x=14$, so $x=2$. Substitute into $y = 5 - 2x$ to get $y=5-2(2)=1$. The solution is $(2, 1)$.

Explanation

This is a step-by-step recipe for success. First, get your equations into $Ax+By=C$ form. Next, multiply to create opposite terms. Then, add, solve for one variable, substitute back to find the other, and always check your answer.

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.