Elimination by Multiplying One Equation
Solve linear systems using elimination by multiplying one equation: scale coefficients to match, add or subtract equations to cancel a variable, then solve and back-substitute.
Key Concepts
Property If adding or subtracting the equations directly does not eliminate a variable, you must alter them first. When one coefficient is a simple multiple of the other, you only need to multiply ONE equation by a constant. By multiplying every single term on both sides of that equation by the chosen constant, you create an equivalent equation with the exact opposite coefficient needed for elimination.
Examples Example 1 (Multiplying One Equation): Solve the system $3x + y = 5$ and $2x 3y = 7$. The $y$ coefficients are $1$ and $ 3$. To make them opposites, multiply the ENTIRE top equation by $3$: $3(3x + y) = 3(5) \rightarrow 9x + 3y = 15$ Now, add this new equation to the bottom equation: $(9x + 2x) + (3y 3y) = 15 + 7$ $11x = 22 \rightarrow x = 2$ Example 2 (Back Substitution): Now that $x = 2$, substitute it back into the original, simplest equation ($3x + y = 5$): $3(2) + y = 5 \rightarrow 6 + y = 5 \rightarrow y = 1$. The solution is $(2, 1)$.
Explanation You can think of an equation like a recipe. If a recipe makes 1 batch of cookies, multiplying every single ingredient by 3 gives you 3 batches, but it is still the exact same recipe! In algebra, multiplying the left and right sides by the same number keeps the line exactly the same, but it changes the numbers to fit your needs. The most common mistake is multiplying the variables but forgetting to multiply the constant on the other side of the equal sign.
Common Questions
When do you need to multiply one equation before using elimination?
You multiply one equation when the coefficients of a chosen variable are not already opposites. Multiply by a constant that makes one pair of coefficients match in absolute value, then add or subtract to eliminate that variable.
What are the steps for elimination by multiplying one equation?
Choose the variable to eliminate. Multiply one equation by the necessary constant. Add or subtract the two equations to cancel that variable. Solve the resulting single-variable equation, then substitute back to find the other variable.
How do you check a solution from elimination?
Substitute both values back into both original equations. If each equation is satisfied, the solution is correct. For a consistent system there should be exactly one solution that satisfies all constraints.