Solving systems by elimination is a method for finding the ordered pair that makes two linear equations true at the same time. This cheat sheet helps students organize each step so the variables line up, one variable cancels, and the remaining equation is easy to solve. It is especially useful when the equations are already in standard form, such as .
The main idea is to add or subtract equivalent equations so one variable has coefficients that are opposites, such as and . If the coefficients are not opposites, multiply one or both equations by constants to create a matching pair. After finding one variable, substitute that value into either original equation, solve for the other variable, and check the ordered pair in both equations.
Key Facts
- A system of two linear equations has a solution that makes both equations true.
- Elimination works best when equations are written in standard form with like terms aligned.
- To eliminate a variable, create opposite coefficients such as and , then add the equations.
- Multiplying an equation by a nonzero constant, such as , creates an equivalent equation with the same solutions.
- If the -coefficients match exactly, subtract the equations to eliminate because .
- If the coefficients are not already matched, use the least common multiple, such as , to choose helpful multipliers.
- After solving for one variable, substitute into an original equation, such as , to find the other variable.
- Always check the final ordered pair by substituting it into both original equations.
Vocabulary
- System of equations
- A set of two or more equations that are solved together using the same variables.
- Elimination
- A solving method that adds or subtracts equations to cancel one variable.
- Coefficient
- The number multiplying a variable, such as in .
- Equivalent equation
- An equation made by valid operations that has the same solutions as the original equation.
- Ordered pair
- A solution written as where the first number is the -value and the second number is the -value.
- Check
- The step of substituting the solution into the original equations to confirm both statements are true.
Common Mistakes to Avoid
- Forgetting to multiply every term in an equation is wrong because means , not .
- Adding when coefficients are the same instead of subtracting is wrong because , so the variable does not cancel.
- Changing only one side of an equation is wrong because it destroys equality and creates a different solution set.
- Substituting into a multiplied equation and then reporting the answer without checking can be risky because arithmetic errors are easier to miss.
- Writing the answer as is wrong because ordered pairs must be written in the order .
Practice Questions
- 1 Solve by elimination: .
- 2 Solve by elimination: .
- 3 Solve by elimination after choosing multipliers: .
- 4 Explain why multiplying one equation by before adding can help solve a system by elimination.
Understanding Solving Systems by Elimination Step by Step
Elimination is based on a useful balance rule. An equation says that two quantities have the same value. Multiplying every term on both sides by the same nonzero number preserves that balance.
This is why a changed-looking equation can still represent the same line. Students sometimes multiply only one term, such as changing three x plus y equals eight into six x plus y equals sixteen. That changes the equation and usually changes its solutions.
Distribute the multiplier to every term, including the constant on the right. Negative multipliers need special care because every sign changes.
Choosing which variable to eliminate can make the work much shorter. Look for coefficients that already match or are easy multiples. For example, coefficients of two and six are often easier to handle than coefficients of seven and nine.
The least common multiple gives a target that both coefficients can reach, but it is not always necessary to use the smallest possible target. A negative multiplier may create opposite coefficients in one step.
Write the multiplied equation on a new line before combining it with the other equation. This keeps the original equations visible for checking later and prevents lost signs.
After equations are combined, the result should be a one-variable equation. Combine like terms carefully, then undo operations in reverse order. The value found first is not the final answer by itself.
It is only one coordinate. Substitution finds the missing coordinate, but using the original equations is safer than using a rewritten equation with many arithmetic steps. Then place the values in the correct order.
An x value of four and a y value of negative two must be recorded as the ordered pair four, negative two. Reversing the order gives a different point.
Sometimes elimination produces a surprising statement instead of a variable value. A true statement such as zero equals zero means the two equations describe the same line. There are infinitely many solution points.
A false statement such as zero equals five means the lines are parallel and never meet, so there is no solution. These outcomes are not mistakes if the arithmetic is correct. Systems appear in real situations when two rules describe the same quantities, such as comparing phone plans with different fees and rates, or finding when two moving objects have the same position.
The graph gives a visual check. One intersection means one solution, overlapping lines mean infinitely many solutions, and parallel lines mean no solution. Checking in both original equations catches arithmetic errors and confirms the result fits the situation.