Solving systems by substitution is a method for finding the values that make two equations true at the same time. It is especially useful when one equation is already solved for a variable, such as y = 2x + 1. The main idea is to replace one variable with an equal expression from the other equation.
This turns a two-variable system into a one-variable equation that can be solved step by step.
The substitution method works because equal quantities can replace each other without changing the truth of an equation. After isolating one variable, you substitute its expression into the other equation, solve for the remaining variable, and then use that value to find the second variable. A final check in both original equations helps confirm that the ordered pair is the solution.
This method connects algebraic reasoning with the graphing idea that a system's solution is the intersection point of two lines.
Key Facts
- A system solution is an ordered pair (x, y) that satisfies both equations.
- If y = expression, substitute that expression for y in the other equation.
- Example substitution: if y = 2x + 3 and x + y = 18, then x + (2x + 3) = 18.
- After substituting, solve the one-variable equation using inverse operations.
- Once one variable is found, plug it back into either original equation to find the other variable.
- Check by substituting the ordered pair into both original equations.
Vocabulary
- System of equations
- A set of two or more equations that use the same variables and are considered together.
- Substitution
- A solving method where an expression equal to one variable is inserted into another equation.
- Isolate
- To rearrange an equation so that a chosen variable is alone on one side.
- Ordered pair
- A pair of numbers written as (x, y) that gives the values of two variables.
- Check
- To test a proposed solution by placing its values into the original equations.
Common Mistakes to Avoid
- Substituting into the same equation you used to isolate the variable, which often leads to an identity instead of a solution. Always substitute into the other equation.
- Dropping parentheses around a substituted expression, which can change signs or operations. Write the replacement in parentheses before simplifying.
- Solving for one variable and forgetting to find the other, which gives only half of the ordered pair. Use the first value in either original equation to find the second value.
- Skipping the check step, which can hide arithmetic or sign errors. Substitute the final ordered pair into both original equations to verify the solution.
Practice Questions
- 1 Solve by substitution: y = 3x - 2 and x + y = 10.
- 2 Solve by substitution: x = 2y + 1 and 3x - y = 14.
- 3 Explain why the substitution method is often easiest when one equation is already solved for x or y.