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.
Understanding Math: Solving Systems by Elimination
The key rule in elimination is that an equation must stay balanced. If you multiply an equation by a number, multiply every term on both sides. For example, if one equation is two x plus y equals seven, multiplying by negative three gives negative six x minus three y equals negative twenty one.
A common mistake is multiplying only the variable terms and forgetting the constant. That creates a different line, so the final answer will be wrong.
Write each changed equation on a new line. This makes the operation visible and reduces sign errors.
Choose the variable that takes the least work to remove. Sometimes one pair of coefficients already has opposite signs. In other cases, find a common multiple.
If the x coefficients are two and three, six is a useful target. Multiply the first equation by three and the second by negative two. Their x terms then cancel when the equations are added.
You can subtract instead when matching coefficients have the same sign. Subtracting one equation means changing the sign of every term in that equation before combining like terms. Many students find it safer to multiply by negative one first, then add.
The result after elimination can reveal more than one ordinary solution. A statement such as zero equals five is impossible. It means the lines are parallel and never meet, so the system has no solution.
A statement such as zero equals zero is always true. It means the two original equations describe the same line, so there are infinitely many solutions. These cases often happen after one equation is a multiple of the other.
Do not try to solve for a variable after getting either kind of zero statement. State what it means about the system instead.
Elimination is useful when two totals describe the same situation. Suppose a school event sold adult tickets and student tickets. One equation can represent the number of tickets sold.
Another can represent the money collected. The variables stand for quantities, and the coefficients come from prices or counts. Before calculating, define each variable clearly and check that units match.
Ticket counts should not be added to dollar amounts. After finding an ordered pair, test both values in both original equations.
This final check catches arithmetic slips and confirms that the answer fits the story. With practice, focus on signs, complete multiplication, and careful combination of like terms.
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.