📚 Topic Summary
Solving systems of equations by substitution is a powerful technique in algebra. When one of the equations already has a variable isolated (e.g., $y = 3x + 2$), the substitution method becomes straightforward. You simply substitute the expression for the isolated variable into the other equation. This reduces the system to a single equation with one variable, which you can then solve. After finding the value of that variable, substitute it back into either original equation to find the value of the other variable. This gives you the solution to the system as an ordered pair $(x, y)$.
For example, consider the system:
$y = x + 1$
$2x + y = 4$
Since $y$ is already isolated in the first equation, substitute $x + 1$ for $y$ in the second equation: $2x + (x + 1) = 4$. Solve for $x$, and then substitute the value of $x$ back into $y = x + 1$ to find $y$.
🧮 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. System of Equations |
A. A method for solving systems of equations by solving for one variable and substituting that expression into the other equation. |
| 2. Substitution |
B. A set of two or more equations containing the same variables. |
| 3. Isolated Variable |
C. A pair of values (x, y) that satisfy both equations in a system. |
| 4. Solution |
D. A variable that is expressed alone on one side of an equation. |
| 5. Ordered Pair |
E. A set of numbers written in the form (x, y) showing the position of a point on the Cartesian plane. |
✍️ Part B: Fill in the Blanks
When solving a system of equations by substitution where a variable is already __________, you can directly __________ the expression into the other equation. This creates a single equation with one __________, which can then be solved. After finding the value of that variable, substitute it back into either original equation to find the value of the other __________. The solution is represented as an __________ pair.
🤔 Part C: Critical Thinking
Explain in your own words why solving for an isolated variable first simplifies the process of solving a system of equations by substitution. Give an example to illustrate your point.