Solve Linear-Quadratic Systems by Elimination
Solving linear-quadratic systems by elimination is a Grade 11 Algebra 1 technique from enVision Chapter 9 where one variable is eliminated to reduce the system to a single-variable quadratic. The 6-step process: write both equations in standard form, create opposite coefficients for one variable, add or subtract to eliminate it, solve the resulting equation, substitute back to find the other variable, and verify as ordered pairs. For example, in the system y = x² - 4 and 2x + y = 2, substituting gives x² + 2x - 6 = 0. Solutions may be 0, 1, or 2 ordered pairs.
Key Concepts
To solve a linear quadratic system by elimination:.
Step 1. Write both equations in standard form. Step 2. If needed, multiply one or both equations so that the coefficients of one variable are opposites. Step 3. Add or subtract the equations to eliminate one variable. Step 4. Solve the resulting equation for the remaining variable. Step 5. Substitute each solution back into either original equation to find the other variable. Step 6. Write the solutions as ordered pairs and check in both original equations.
Common Questions
What are the steps to solve a linear-quadratic system by elimination?
Write both equations in standard form, multiply to create opposite coefficients, add or subtract to eliminate one variable, solve the resulting quadratic, substitute back into the linear equation, and write solutions as ordered pairs.
When is elimination the best method for linear-quadratic systems?
When a linear variable can be easily eliminated because coefficients are already opposites or can be made opposites with simple multiplication.
How do you solve the system x² + y² = 25 and x + y = 7 by elimination?
Solve the linear equation for y: y = 7 - x. Substitute into x² + (7-x)² = 25 and expand. The solutions are (3, 4) and (4, 3).
How many solutions can a linear-quadratic system have?
Zero, one, or two solutions. The number depends on how many times the line intersects the parabola or curve.
What happens after you eliminate a variable in a linear-quadratic system?
You get a single-variable quadratic equation that you solve using factoring, the quadratic formula, or completing the square.
Do you always need to check solutions in both original equations?
Yes. Each solution found must be substituted back into both original equations to confirm it satisfies the entire system.