Use substitution to form and solve a quadratic equation in order to solve a related equation

Equations, Inequalities and Graphs

1. What is substitution? 🤔

Think of substitution like swapping a character in a story. If you know that y = 3x + 2 in one part of the story, you can replace every y in another part with that expression. The story (equation) stays the same, but it’s now written only in terms of x.

2. Forming a quadratic equation by substitution

Let’s walk through a typical example:

1. Start with a system of two equations:
   $x + y = 5$ and $xy = 6$.
2. Choose one equation to solve for one variable. From the first, $y = 5 - x$.
3. Substitute this expression into the second equation:
   $x(5 - x) = 6$.
4. Expand and rearrange to get a standard quadratic form:
   $-x^2 + 5x - 6 = 0$ → multiply by $-1$ to make it easier: $x^2 - 5x + 6 = 0$.

3. Solving the quadratic equation ??

Now we solve $x^2 - 5x + 6 = 0$ using one of the following methods:

• Factoring: Find two numbers that multiply to $6$ and add to $-5$. They are $-2$ and $-3$. So, $(x-2)(x-3)=0$.
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ with $a=1, b=-5, c=6$.
• Completing the square: Add and subtract the square of half the coefficient of $x$.

Both methods give the solutions $x = 2$ and $x = 3$.

4. Applying the solution back to the original equation 📚

Remember we had $y = 5 - x$. Plug each $x$ value back in:

1. $x = 2$ → $y = 5 - 2 = 3$
2. $x = 3$ → $y = 5 - 3 = 2$

So the solutions to the original system are $(2,3)$ and $(3,2)$.

5. Common pitfalls and exam tips ⚠️