Solved examples: Linear-quadratic systems of equations (Step-by-step)

📚 Quick Study Guide

• 📈 A linear equation is an equation that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• 🧮 A quadratic equation is an equation that can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• 💡 A system of equations consists of two or more equations that are solved simultaneously.
• 🎯 To solve a linear-quadratic system, you typically use substitution. Solve the linear equation for one variable, then substitute that expression into the quadratic equation.
• ➗ After substitution, you'll have a quadratic equation in one variable. Solve this equation using factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• ✍️ Once you have the $x$-value(s), substitute them back into the linear equation to find the corresponding $y$-value(s).
• ✅ The solutions are the points where the line and parabola intersect. These are written as ordered pairs $(x, y)$.

🧪 Practice Quiz

1. Question 1: What is the first step in solving a linear-quadratic system of equations using substitution?
   1. Solve the quadratic equation for $x$.
   2. Solve the linear equation for one variable.
   3. Set the two equations equal to each other.
   4. Graph both equations.
2. Question 2: Given the system $y = x + 1$ and $y = x^2 - 2x + 1$, what is the result of substituting the linear equation into the quadratic equation?
   1. $x + 1 = x^2 - 2x + 1$
   2. $x - 1 = x^2 + 2x - 1$
   3. $y + 1 = y^2 - 2y + 1$
   4. $y = x^2 - 2x + 1$
3. Question 3: After substituting and simplifying, you arrive at the quadratic equation $x^2 - 3x = 0$. What are the solutions for $x$?
   1. $x = 0, 3$
   2. $x = 1, 2$
   3. $x = -3, 0$
   4. $x = -1, -2$
4. Question 4: If $x = 0$ is a solution to the system and $y = x + 1$, what is the corresponding $y$-value?
   1. $y = 0$
   2. $y = 1$
   3. $y = -1$
   4. $y = 2$
5. Question 5: If $x = 3$ is a solution to the system and $y = x + 1$, what is the corresponding $y$-value?
   1. $y = 2$
   2. $y = 3$
   3. $y = 4$
   4. $y = 5$
6. Question 6: How many possible solutions can a linear-quadratic system have?
   1. 0, 1, or 2
   2. Only 1
   3. Only 2
   4. Infinitely many
7. Question 7: Which method is generally preferred for solving linear-quadratic systems?
   1. Elimination
   2. Graphing
   3. Substitution
   4. Cross-multiplication