1. **Understanding Systems of Linear Equations** A system of linear equations is a set of two or more equations with the same variables. Our goal is to find values for these variables that make all equations true at the same time. 2. **Recognizing the Type of Problem** Look for equations with variables like $x$ and $y$ that are linear (no exponents or powers). For example, $x + y = 5$ and $2x - y = 3$ form a system. 3. **Methods to Solve Systems** There are three common methods: - **Graphing:** Plot both equations on a graph and find where they intersect. - **Substitution:** Solve one equation for one variable, then substitute that into the other. - **Elimination:** Add or subtract equations to eliminate one variable, making it easier to solve. 4. **Step-by-Step Example Using Substitution** Solve the system: $$\begin{cases} x + 4y = 6 \\ 2x - 3y = 1 \end{cases}$$ **Step 1:** Solve the first equation for $x$: $$x = 6 - 4y$$ *Why?* We want to express $x$ in terms of $y$ to substitute into the second equation. **Step 2:** Substitute $x = 6 - 4y$ into the second equation: $$2(6 - 4y) - 3y = 1$$ *Why?* This replaces $x$ with an expression involving only $y$, so we can solve for $y$. **Step 3:** Simplify and solve for $y$: $$12 - 8y - 3y = 1$$ $$12 - 11y = 1$$ Subtract 12 from both sides: $$\cancel{12} - 11y = 1 - \cancel{12}$$ $$-11y = -11$$ Divide both sides by $-11$: $$y = \frac{-11}{-11} = 1$$ *Why?* We isolate $y$ to find its value. **Step 4:** Substitute $y=1$ back into $x = 6 - 4y$: $$x = 6 - 4(1) = 6 - 4 = 2$$ *Why?* To find the corresponding $x$ value. **Answer:** $x=2$, $y=1$. 5. **Tips to Remember** - Always isolate one variable first. - Substitute carefully to avoid mistakes. - Check your solution by plugging values back into original equations. 6. **Your Turn!** Try solving this system using substitution: $$\begin{cases} y = 6 - 3x \\ y = 2x + 1 \end{cases}$$ *Hint:* Since both expressions equal $y$, set them equal to each other and solve for $x$. If you get stuck, ask me for help!