Solving systems by elimination is a powerful method for finding the point where two linear equations are true at the same time. It matters because many real situations involve two unknown quantities, such as ticket prices, mixture amounts, or speeds. Elimination works especially well when the equations are already in standard form, such as ax + by = c.
The goal is to remove one variable so the other variable can be solved directly.
To eliminate a variable, you may need to multiply one or both equations so the coefficients of one variable become opposites. Then you add the equations, causing that variable to cancel and leaving a one-variable equation. After solving for one variable, you substitute that value into one of the original equations to find the other variable.
The final answer is written as an ordered pair, which represents the intersection point of the two lines.
Key Facts
- A system of linear equations has a solution where both equations are true at the same time.
- Standard form is ax + by = c, which is often convenient for elimination.
- To eliminate x, make the x-coefficients opposites, such as 3x and -3x.
- To eliminate y, make the y-coefficients opposites, such as 5y and -5y.
- If 2x + 3y = 13 and 4x - 3y = 5, then adding gives 6x = 18, so x = 3.
- After finding one variable, use substitution into an original equation to find the other variable.
Vocabulary
- System of equations
- A set of two or more equations that use the same variables and are solved together.
- Elimination
- A method for solving a system by adding or subtracting equations to cancel one variable.
- Coefficient
- The number multiplying a variable, such as 4 in 4x.
- Back-substitution
- The step of substituting a solved variable value into an equation to find the remaining variable.
- Ordered pair
- A solution written as (x, y) that gives the values of both variables.
Common Mistakes to Avoid
- Adding only one side of the equations is wrong because equality must be maintained on both sides. Always combine the left sides together and the right sides together.
- Forgetting to multiply every term when scaling an equation is wrong because it changes the equation unevenly. If you multiply an equation by 3, multiply all terms on both sides by 3.
- Canceling variables with the same sign is wrong because 4x + 4x equals 8x, not 0. The coefficients must be opposites, such as 4x and -4x.
- Stopping after finding one variable is incomplete because a system solution needs both x and y. Always back-substitute and write the final answer as an ordered pair.
Practice Questions
- 1 Solve by elimination: 2x + 3y = 16 and 4x - 3y = 8.
- 2 Solve by elimination: 5x + 2y = 1 and 3x - 2y = 15.
- 3 Explain why multiplying one equation by a constant before adding does not change the solution of the system, as long as every term in that equation is multiplied.