91Ó°ÊÓ

Quadratic equations play a big role in math. They describe curves on a graph called parabolas. The standard form is: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Understanding how to solve these is key. For example, the equation \(2y^2 - 7y + 5 = 0\) seen in the original exercise comes from rearranging terms.
We see these equations in many real-world contexts, like physics or finance. Learning about them helps you solve a wide range of problems.

The substitution method is a powerful tool for solving systems of equations. This method involves isolating one variable from one equation and substituting its value into the other. Let’s break it down. In our exercise:
1. First, isolate \(x\) from the second equation: \(4y + x = 7\)
2. Solve for \(x\) and get \(x = 7 - 4y\).
3. Next, substitute this value into the first equation: \(2y^2 + xy = 5\).
4. Replace \(x\) in the first equation: \(2y^2 + y(7 - 4y) = 5\).
5. Simplify and solve the quadratic equation for \(y\).
By substituting one variable into the other equation, you reduce a system of equations to a single equation with one variable, making it easier to solve.
This method is very useful when it’s hard to solve equations directly.

Solving a quadratic equation can be tricky. But the quadratic formula makes it simpler. This formula is: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Let’s use this in our exercise’s quadratic equation: \(2y^2 - 7y + 5 = 0\).
Here’s how to do it:

• Identify values: \(a = 2\), \(b = -7\), \(c = 5\).
• Put them into the formula:
  \(y = \frac{7 \pm \sqrt{49 - 40}}{4}\).
• Simplify:
  \(y = \frac{7 \pm 3}{4}\).

This results in two values: \(y = 2.5\) and \(y = 1\). Using the quadratic formula is vital because it works for any quadratic equation, giving you accurate solutions quickly.

Graphing equations helps visualize their solutions. In our exercise, confirming solutions with a graph is mentioned. Here's how graphs can help:
- Each equation represents a line or curve
- The solutions to the system are where the lines or curves intersect.
In our case:

• The first equation \(2y^2 + xy = 5\) graphs a curve.
• The second equation \(4y + x = 7\) graphs a straight line.

You can plot these on a graph. The points where they meet are the solutions. For this exercise, they intersect at \((-3, 2.5)\) and \((3, 1)\). Using a graphing tool or calculator can confirm the algebraic solutions visually, making it easier to understand.