91Ó°ÊÓ

In math, an ordered pair \((x, y)\) is a pair of numbers used to locate a point on a coordinate plane. The first number is the x-coordinate, and the second one is the y-coordinate. Ordered pairs are often used to solve problems where you need to find the values for variables that satisfy a given equation.
For example, consider the exercise with the equation \[3x - 5y = 10\]. The task is to find suitable ordered pairs that satisfy this equation. This means that we must find values for either x or y that will make the equation true.
To understand this better, always follow these steps:

• Identify the given values of x and y.
• Substitute the given value into the equation.
• Solve for the unknown variable to complete the ordered pair.

This method provides a straightforward way to discover ordered pairs that satisfy specific equations.

The substitution method is a widely used technique to solve equations, especially when working with ordered pairs. It involves replacing a variable with its equivalent value or expression to simplify the equation.
Let’s use the example from our exercise:

• Given the equation \[3x - 5y = 10\], suppose we know that \(x = -2/3\).
• We replace x in the equation with \(-2/3\).

Here’s how it works step-by-step:

• Starting equation: \[3x - 5y = 10\]
• Substitute \(x = -2/3\): \[3(-2/3) - 5y = 10\]
• Simplify: \[-2 - 5y = 10\]
• Solve for y:
  \[-5y = 12\]
  \(y = -12/5\)

Through substitution, we were able to find that when \(x = -2/3\), \((y = -12/5)\). This process can easily be applied to find missing values in other ordered pairs.

Algebraic manipulation refers to using algebraic methods to simplify or rearrange equations to solve for unknown variables. It’s a vital skill in solving linear equations.
Take, for instance, solving for y when \(x = 5\) in the given equation \[3x - 5y = 10\]:

• Starting equation: \[3(5) - 5y = 10\]
• Simplify: \[15 - 5y = 10\]
• Subtract 15 from both sides: \[-5y = -5\]
• Divide by -5: \[( y = 1)\]

Essentially, you isolate the variable you are solving for through a series of steps: addition, subtraction, multiplication, or division. Algebraic manipulation helps in maintaining the balance of equations while moving terms across the equal sign.
This skill is basically operating the variables and constants through various arithmetic operations to find the desired solution. Learning algebraic manipulation makes handling complex equations a lot simpler and more intuitive.