Solving Systems of Equations By Elimination & Substitution With 2 Variables

Solving Systems of Equations: Elimination Method
📌 Solved the system $2x + 3y = 8$ and $5x - 3y = -1$ by adding the equations to eliminate $y$, resulting in $7x = 7$, thus $x=1$.
➕ Substituting $x=1$ into the first equation yielded $2(1) + 3y = 8$, simplifying to $3y = 6$, so $y=2$. The solution is the ordered pair (1, 2).
➗ For $2x + 5y = 19$ and $x - 2y = -4$, the second equation was multiplied by -2 to cancel $x$, resulting in $9y = 27$, so $y=3$.
✅ Substituting $y=3$ gave $x - 2(3) = -4$, leading to $x=2$. The solution is (2, 3).

Solving Systems of Equations: Substitution Method
📌 For $y = 5 - 2x$ and $4x + 3y = 13$, $y$ was substituted into the second equation, leading to $4x + 3(5 - 2x) = 13$.
🔢 Simplifying resulted in $-2x + 15 = 13$, which gives $-2x = -2$, so $x=1$.
➗ Plugging $x=1$ back into $y = 5 - 2x$ gives $y = 5 - 2(1) = 3$. The solution is (1, 3).
↔️ When both equations are given as $y$, set them equal: $3x + 2 = 7x - 6$. This yields $4x = 8$, meaning $x=2$.
✔️ Substituting $x=2$ gives $y = 3(2) + 2 = 8$. The solution is (2, 8).

Substitution Method with Rearrangement
📌 For $4x + 2y = 14$ and $3x - 5y = -22$, the first equation was rearranged to solve for $y$: $y = -2x + 7$.
➕ This expression for $y$ was substituted into the second equation: $3x - 5(-2x + 7) = -22$.
✖️ Distributing $-5$ resulted in $3x + 10x - 35 = -22$, combining terms to get $13x = 13$, so $x=1$.
🧩 Substituting $x=1$ into $y = -2x + 7$ yields $y = -2(1) + 7 = 5$. The solution is (1, 5).

Key Points & Insights
➡️ The Elimination Method works best when variables have opposite coefficients (like $3y$ and $-3y$); simply add the two equations.
➡️ If coefficients do not cancel, use multiplication to create opposite coefficients (e.g., multiply by -2 to cancel $2x$).
➡️ The Substitution Method is effective when one variable is already isolated (e.g., $y=...$ or $x=...$).
➡️ When using substitution, always rearrange the equation first if necessary (by isolating $x$ or $y$) before plugging it into the other equation.

📸 Video summarized with SummaryTube.com on Mar 09, 2026, 03:36 UTC