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Chapter 5: Problem 1

(4.1) Determine if each ordered pair is a solution of the given equation. $$4 x-y=9 ;(1,-5)$$

Short Answer

Expert verified
Yes, the ordered pair (1, -5) is a solution of the given equation \(4x - y = 9\), since substituting the values results in a true statement: \(9 = 9\).

Step by step solution

01

Identify the x and y values from the ordered pair

We're given the ordered pair (1, -5), so we can see that the x-value is 1 and the y-value is -5.
02

Plug the x and y values into the equation

Now, we'll plug in the x and y values we found in step 1 into the given equation: \(4x - y = 9\). Replace x with 1 and y with -5: $$4(1) - (-5) = 9$$
03

Simplify and check the equation

Next, let's compute the expression and check if it is valid: $$4(1) - (-5) = 9 \Rightarrow 4 + 5 = 9$$ The equation simplifies to $$9 = 9$$
04

Determine if the ordered pair is a solution

Finally, we can see that substituting the ordered pair (1, -5) into the equation gives us a true statement. Therefore, the ordered pair (1, -5) is a solution of the given equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
An ordered pair is a fundamental concept in mathematics presented as \((x, y)\).These pairs are often used in equations and graphs to denote specific values of \(x\) and \(y\).The sequence matters because the first element is always the \(x\)-value and the second element is the \(y\)-value.
  • x-value: The horizontal value in a coordinate pair, often representing a variable in an equation.
  • y-value: The vertical value, which often links to another variable in the equation.
In our exercise, the ordered pair \((1, -5)\) means \(x = 1\) and \(y = -5\).By correctly identifying these values, you can easily test if they satisfy a particular equation.Understanding ordered pairs lays the foundation for learning how to plot points on a coordinate plane and to solve equations.
Equation Substitution
Equation substitution refers to the process of replacing the variables in an equation with specific numbers to determine if those numbers satisfy the equation.In the given exercise, we have the equation \(4x - y = 9\).By substituting the ordered pair \((1, -5)\), we replace \(x\) with 1 and \(y\) with -5.

Here's how it looks:
  • Replace \(x\) with 1: \(4(1)\)
  • Replace \(y\) with -5: \(-(-5)\)
So the equation becomes\[4(1) - (-5) = 9\].Substitute \(x\) and \(y\) accurately to test if the equation holds true.This process helps in verifying whether an ordered pair is a solution for the equation or not.
Checking Solutions
Checking solutions involves verifying if the simplified equation after substitution leads to a true statement.If it does, the ordered pair is indeed a solution to the given equation.In our exercise:
  • Calculate \(4(1) - (-5)\), which simplifies to \[4 + 5 = 9\].
By confirming that\(9 = 9\),we establish that the ordered pair \((1, -5)\) satisfies the equation \(4x - y = 9\).
The process of checking solutions ensures accuracy by confirming the left side of the equation equals the right side.This step reassures whether the chosen values for \(x\) and \(y\) correctly satisfy the equation, validating the solution.

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