6-3 Elimination Using Addition and Subtraction

Property

The elimination method is an algebraic technique used to solve a system of equations by adding them together to eliminate one variable.
This method is highly efficient when the two equations are in standard form ($Ax + By = C$) and one of the variables has opposite coefficients (e.g., $2y$ and $-2y$).
Once the equations are added and a variable is eliminated, you solve for the remaining variable. Finally, use back-substitution: plug that value back into either of the original equations to find the second variable.

Examples

• Example 1 (Adding to Eliminate): Solve the system $x + y = 8$ and $x - y = 2$.

Notice that the $y$ coefficients are $1$ and $-1$. Add the equations vertically:
$(x + x) + (y - y) = 8 + 2$
$2x = 10 \rightarrow x = 5$

• Example 2 (Back-Substitution): Now that we know $x = 5$, substitute it back into the first equation ($x + y = 8$).

$5 + y = 8 \rightarrow y = 3$. The solution is the ordered pair $(5, 3)$.

• Example 3: Solve $3x + 2y = 16$ and $-x - 2y = -8$.

Add vertically: $(3x - x) + (2y - 2y) = 16 - 8$
$2x = 8 \rightarrow x = 4$.
Back-substitute into the first equation: $3(4) + 2y = 16 \rightarrow 12 + 2y = 16 \rightarrow 2y = 4 \rightarrow y = 2$. Solution: $(4, 2)$.

Explanation

Property

When a variable in both equations has the exact same coefficient (e.g., $4x$ and $4x$), adding the equations will not eliminate it. Instead, you must subtract the entire second equation from the first.

To do this correctly, you must distribute the negative sign to every single term in the bottom equation (changing all their signs) and then add the equations together.

Examples

• Distributing the Negative: Subtract $(4x - 7)$ from $(6x + 2)$.

Write it out: $(6x + 2) - (4x - 7)$.
Distribute the minus sign to flip the signs inside: $6x + 2 - 4x + 7$.
Combine like terms: $2x + 9$.

• Subtracting Equations: Solve $5x + 3y = 17$ and $2x + 3y = 8$.

Since the $y$ terms are identical ($3y$), subtract the entire bottom equation:
$-(2x + 3y = 8) \rightarrow -2x - 3y = -8$
Now add this to the top equation:
$(5x - 2x) + (3y - 3y) = 17 - 8$
$3x = 9 \rightarrow x = 3$.
Back-substitute: $2(3) + 3y = 8 \rightarrow 6 + 3y = 8 \rightarrow 3y = 2 \rightarrow y = \frac{2}{3}$.

Explanation

Subtracting an entire equation is exactly the same as multiplying it by -1 and then adding. The most common mistake in algebra is subtracting the first term but forgetting to subtract the rest! To avoid this trap, do not try to subtract in your head. Physically draw parentheses around the bottom equation, write a minus sign outside, and rewrite the equation with every single sign flipped. Then, just add them normally.

Property

Sometimes, applying the elimination method causes both variables to cancel out at the same time. The resulting numerical statement reveals the classification of the system:

• Dependent System (Infinite Solutions): If the result is a true statement like $0 = 0$, the equations describe the exact same line.
• Inconsistent System (No Solution): If the result is a false statement like $0 = 7$, the equations describe parallel lines that never intersect.

Examples

• Dependent System: Add the equations $2x + y = 4$ and $-2x - y = -4$.

Adding vertically gives: $0x + 0y = 0 \rightarrow 0 = 0$.
This is always true, meaning there are infinitely many solutions.

• Inconsistent System: Add the equations $3x - y = 5$ and $-3x + y = 2$.

Adding vertically gives: $0x + 0y = 7 \rightarrow 0 = 7$.
This is a mathematical contradiction (false), meaning there is no solution.

• Common Error: When elimination gives $0 = 0$, do not write the ordered pair $(0, 0)$ as the solution. $(0, 0)$ is a specific point on the graph, whereas $0 = 0$ means every point on the line is a solution.

Explanation

When both variables vanish into thin air, your algebra is acting as an early warning system about the geometry of the lines. A true statement ($0=0$) means the left side of your equations and the right side of your equations were perfectly proportional the whole time—they are identical overlapping lines! A false statement ($0=7$) means the lines have the same slope but different spacing, trapping them in parallel lanes forever.