Example Card: Rearranging Before Substitution
Solve systems using substitution in Grade 9 algebra by first rearranging one equation to isolate a variable, then substituting that expression into the second equation to find the solution.
Key Concepts
Sometimes neither equation is ready for substitution. Let's see how to cleverly rearrange one to unlock the solution.
Example Problem Solve the system of equations by substitution: $3x y = 9$ and $x + 2y = 4$.
Step by Step 1. First, we need to rearrange one equation so a variable is isolated. The first equation, $3x y = 9$, can be easily solved for $y$. 2. Start with the first equation: $$3x y = 9$$ 3. Subtract $3x$ from both sides: $$ y = 3x + 9$$ 4. Multiply the entire equation by $ 1$ to solve for $y$: $$y = 3x 9$$ 5. Now, substitute this expression for $y$ into the second equation. $$x + 2y = 4$$ $$x + 2(3x 9) = 4$$ 6. Distribute the $2$ to the terms in the parentheses: $$x + 6x 18 = 4$$ 7. Combine the like terms ($x$ and $6x$): $$7x 18 = 4$$ 8. Add $18$ to both sides to isolate the term with $x$: $$7x = 14$$ 9. Divide by $7$ to solve for $x$: $$x = 2$$ 10. Substitute $x = 2$ back into one of the original equations to find $y$. Let's use the first one: $$3(2) y = 9$$ $$6 y = 9$$ $$ y = 3$$ $$y = 3$$ 11. The solution is the ordered pair $(2, 3)$.
Common Questions
When do you need to rearrange before substitution?
When neither equation already has a variable isolated (like y = ... or x = ...), you must solve one equation for a variable first. Choose the variable with coefficient 1 to avoid fractions. Rearranging first makes substitution cleaner.
How do you solve the system 3x - y = 9 and x + 2y = -4 by rearranging then substituting?
Rearrange the first equation for y: y = 3x - 9. Substitute into the second: x + 2(3x - 9) = -4 → x + 6x - 18 = -4 → 7x = 14 → x = 2. Back-substitute: y = 3(2) - 9 = -3. Solution: (2, -3).
What makes substitution more efficient than elimination for certain systems?
Substitution is more efficient when one equation has a variable with coefficient 1 (or -1), making isolation clean without fractions. Elimination is preferred when coefficients are larger or rearrangement would create fractions.