Substitution and isolation are two mathematical techniques that are closely related and often used together to solve equations or manipulate expressions.
Substitution involves replacing a variable in an equation or expression with a specific value or another expression. This is usually done to simplify the equation or expression and make it easier to solve or manipulate. By substituting a specific value for a variable, we can evaluate the equation or expression for that value.
Isolation, on the other hand, involves rearranging an equation or expression to make a specific variable the subject or isolate it on one side of the equation. This is done to highlight the relationship and properties of that specific variable within the equation.
When solving equations or manipulating expressions, substitution and isolation often go hand in hand. Substitution is usually performed first to simplify the equation or expression, and then isolation is used to rearrange the equation or expression to focus on a specific variable or solve for it.
For example, consider the equation 3x + 2y = 10. If we want to solve for y in terms of x, we can use substitution to simplify the equation. We can substitute the value of x in terms of y from another equation or expression. Let's say x = 2y - 1. By substituting this value in the equation, we get:
3(2y - 1) + 2y = 10
Now, to solve for y, we can use isolation. We can rearrange the equation to isolate the variable y:
6y - 3 + 2y = 10
8y - 3 = 10
8y = 13
y = 13/8
In this example, substitution was used to simplify the equation by replacing x with 2y - 1. Then, isolation was used to rearrange the equation and isolate the variable y on one side. The answer y = 13/8 was obtained as the result of solving the equation using these techniques.
Explain how substitution and isolation is related?
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